One subset of Gnathostomata is Osteichthyes, bony vertebrates
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Cartilaginous Fishes (Chondrichthyes) diverged from Osteichthyes, who's skeletons were heavily calcified into bone.  These again divided between teleosts, the ray-finned vertebrates (Actinopterygii), and lobe-finned vertebrates (Sarcopterygii).  On the issue of cartilaginous skeletons as opposed to those made of bone, I should point out that fossil Sarcopterygiian/Crossopterygiian fish had a significant percentage of cartilage in their bones. This is evident in modern coelacanths, lungfish, gar, and the polypterus. 
Polypteriforms, (another "ancient" fish) still have much more cartilage in her skeleton than any modern (teleost) fish. So it appears that gnathostomes (jawed vertebrates) diverged at some point in the late Silurian or early Devonian period, with chondrychthyes developing progressively more cartilage in their skeletons (except in the jaw of course) while osteichthyes used progressively more calcium except where these hinge together.
fossil polypterus
modern polypterus
One subset of Osteichthyes is Sarcopterygii, bony vertebrates which have both lungs and legs.
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Most of the bony fish in the Devonian period had lobed-fins, meaning fins that were attached to limbs complete with bones. But many of these, even some of the most ancient ones, also had lungs. This is evident on every surviving "ancient" species including polypterus. The development of the lung appears to immediately follow the of divergence of bony fish from sharks, and that lung was nothing more than another sort of birth defect, an asymmetrical distension of the buoyancy bladder common to all bony fish. Polypteriforms and other "snake-headed" fish are obligate air-breathers in their natural environment of warm, shallow, oxygen-depleted waters. And they can even use their short little limbs to amble away from drying ponds across dry land to find larger bodies of water. Other fish, even those without actual lungs, use their buoyancy bladders to supplement their oxygen supply when the water is too warm or stagnant to respire with their gills. This is when you see fish taking "drinks" of air.

Cladists (people who really get into this classification system) tend not to use the word, "fish". Somehow, that word seems to have no meaning for them. They say "chordates" instead. Similarly, Sarcopterygii (also known as Crossopterygii) doesn't just refer some ancient order of fish which are now mostly extinct. It refers to them and everything evidently descended from them. Basically, sarcopterygii generally refers to fish, but really includes all organic RNA/DNA protein-based, metabolic, metazoic, nucleic, diploid, bilaterally-symmetrical, digestive, tryploblast, opisthokont, deuterostome coelemates with spinal chords, skulls, backbones, jaws, lungs, and legs.  So humans are included in this classification. 

Most of the features found on Sarcopterygiian fish are unique to their kind and of course their evident descendants. There are only a couple of things they have which other animals had also, but which weren't inherited from any common ancestor. One of these are eyes. In the story of evolutionary history, it seems that eyes arose independently in three different lines, chordates, mollusks, and arthropods. And this is where we have to talk about homology versus analogy.

If these weren't the product of biological evolution, if they were all magically-created, unrelated to anything else, by a common designer, then they should reflect a common design. Why should cephalopods all have the same kind of eye, while all of the chordates have a different design that is common to all of them as well? How come none of the mollusks happen to have the same type eyes as a chordate? Why is it that none of the chordates have the same type eye as a mollusk?  A supernatural designer could have given any type of organ to any individual species he wanted. But evolution demands that each group share the same fundamental structure as their closest relatives, which they do.

Another analogous feature are the legs, which are also shared by arthropods and [some] mollusks again. But like the eyes, these aren't based on the same sorts of designs at all. The "legs" of cephalopods (octopus and their kin) for example, are more like grossly-overgrown lips than legs, and have neither shell nor bone to them. Arthropods have true legs, but they're constructed completely different from any chordate's legs in virtually every respect except their function. These are analogous features. Homologous features are the things we've been talking about up to this point; features that are fundamentally, structurally, and physiologically identical in either example being discussed, and could easily be inherited one to the other.

The legs of early sarcopterygiians are definitely homologous to our own, and to every other terrestrial vertebrate. As you can see in this illustration, every diverse form is built on the same skeletal pattern, matching each other bone-for-bone, and varying only in the shape or length of those bones. This also goes for things like horses and cattle and three-toed sloths, because we can show fossils from each group to see that each of them used to adhere to this same plan, even if some of those bones were eventually lost in their modern relatives.
One subset of Sarcopterygii are the Stegocephalians; limbed vertebrates with digits on the ends of their appendages.
Chordates are deuterostomes while arthropods are ecdysozoans, and mollusks are lophotrochozoans. The basal forms of all three sister groups are blind. In other words, none of their sighted sub-groups could have inherited vision from each other. But when we examine their eyes, we find that they are fundamentally unique in each case. For example, arthropods, (insects, arachnids, crustaceans, etc.) have a multi-lens compound that is nothing like the eyes of either of the other groups; and usually, they each have more than two of them. Some of the ancient horseshoe crabs had them all over, even down their tails. In most cases however, the only eyes that remain are binocular, and located close to the mouth where they are the most needed.

Cephalapoid mollusks have eyes functionally very similar to our own, and much more advanced than those of most chordates. But this is considered convergent evolution because the fundamental structure of their eyes is quite different from ours, even though the needs of performance demand that they function the same. In some respects, their eyes are of a better original design than ours. For instance, they don't have the faint blind spot in the center of our vision that we do.
Those looking for transitional species have several sequences available in the transition from fins to fingers, beginning with Sauripterus, the first sarcopterygii to show the presence of all the bones listed in the images here.
"One confounding aspect of Sauripterus is the presence of eight finger-like radials in the adult fin. As with fingers, these radials are jointed and six of them articulate with the ulnare and the intermedium; both homologues of the carpus; the other two radials are conspicuously more robust and articulate directly with the radius."
Below is a sculpture I did of Elginerpeton, what I consider to be the real-life animal depicted on all the Darwinfish bumper stickers. I did this piece a few years ago, when I didn't know so much about them. But I had the assistance (via email) of paleoartist, Richard Hammond, and the famous Cambridge professor of vertebrate paleontology, Dr. Jennifer Clack, the world's foremost expert on the fauna of that period.  We don't have enough of the skeleton to be certain about the whole shape. But this is one of several contemporary species in this apparent sequence, and they all look about the same. So I've made this one consistent with its siblings. Notice that both dorsal fins and the anal fin are absent from this rendering, as they are on all similar species from this class. And as is typical, the tail fin is not fluked. The first fish with fluked tails lived alongside this one. But their tails are still obviously quite primitive, like a salamander's tail. Most sarcopterygiian fish never had a more fish-like tail than Elginerpeton did.
And as you can see, apart from the broad, internal gills, (which sadly didn't show up well in this photo) this fish already looks very much like some salamanders. And in fact it remains a fish for only two reasons, (1) it still has complete internal gills in addition to its lungs, and (2) its skeletal structure is such that it cannot adequately support its weight outside of water. It could flop and drag itself along perhaps. But it couldn't walk or even crawl unless it did so along the bottom, which it very likely did. It likely didn't have a pelvis yet. But the pelvic fins already have bones and toes in them as well. We don't know how many digits Elginerpeton had either, so I gave him seven on the front fin and eight on the back. This is consistent with the trend seen in this particular subclass.
Some of early stegocephalians, like Livonia multidentada, are known only from skeletal fragments that are matched by homology, and found to be exactly half-fish / half-amphibian. So once again, those seeking transitional species can delight in each of the examples listed on this page. Most fossils are fragmentary. That's just the nature of nature. But many of them, like those listed below, are still complete enough to know for sure exactly what we're looking at.
This Acanthostega gunnari is one of two by Richard Hammond, the paleoartist who assisted me with my own Devonian fish sculpture. Unlike Elginerpeton, this fossil was complete.  Despite the contentions of pseudoscientists, both animals were true fish, and stem-tetrapods (four legged animals) both at the same time.  As such, they're both perfect examples of "transitional species."

Most of those trying to argue against evolution wouldn't even define the term, "transitional species".  In fact they prefer to leave most terms undefined.  This is so they can "move the goal posts" to prevent them from being being accountable whenever they've been proven wrong.  But one troupe of vehement antiscientists actually did post a definition of "transitional" or "intermediate" species, and it was even an accurate one used by evolutionary biologists as well.
One subset of that is Stegocephali is Tetrapoda, gill-less Stegocephalians which are skeletally-adapted for four limbs.
"A transitional fossil is one that looks like it's from an organism intermediate between two lineages, meaning it has some characteristics of lineage A, some characteristics of lineage B, and probably some characteristics part way between the two. Transitional fossils can occur between groups of any taxonomic level, such as between species, between orders, etc. Ideally, the transitional fossil should be found stratigraphically between the first occurrence of the ancestral lineage and the first occurrence of the descendent lineage."
Of course, the creationists who posted this also claimed no transitional species had ever been found.  I presented a list of about 100 perfect matches for all their collective criteria, and they blocked my submissions from their bulletin board rather than be forced to admit that many transitional species do in fact exist, and that they already knew that all along, the very height of intellectual dishonesty.  But I have learned to expect that behavior from their ilk.  So I documented the entire exchange for posterity.
The next important stage in our sequence is Ichthyostega, another tetrapod-fish combo, this time with the tetrapod features dominant, and the fish features vestigial. The tail fin-bars are still there, but they've faded significantly, and will soon be gone. The gills are no longer internal, although it may have switched to distended external gills similar to those of the axolotl. Ichthyostega was still not very adept on land, but it no longer fit the image of a fish-out-of-water either.
The loss of internal gills is one of the dividing lines between what are traditionally and informally known as "fish", and the next important clade in our discussion. There is some debate about this, but Tetrapoda still boils down to mean two things, "stegocephalians lacking internal gills", and a "body plan based on four-limbs", -which these things have obviously already had for some time. But a true tetrapod has also developed the necessary skeletal adaptations to securely mount and operate those four limbs effectively. The explanation for why all terrestrial vertebrates are based on a originally quadrupedal configuration is that they inherited that, and modified it.  This goes for all those vertebrates which were evidently once terrestrial, but returned to the water. Frogs, dogs, hogs, snakes and salamanders, bats and birds, turtles and pterosaurs, pelycosaurs, ichthyosaurs, elasmosaurs, and dinosaurs, lizards, lemurs, llamas, whales, mice and men, are all tetrapods not by some creative coincidence of magical manufacture, but by an obvious natural inheritance.
Once again, specially-created "creatures" wouldn't have to adhere to this structure, which remains plainly homologous even under the most critical examination. Humans have made up lots of vertebrate hexapods in our mythology; Pegasus, griffins, centaurs, and dragons with four legs and two wings, gargoyles, winged angels, Odin's six-legged horse, Sleipnir, and things of this sort. None of these things conform to taxonomy simply because they don't have to. That's the nature of specially-created things. But absolutely everything that really exists does conform to this pattern without deviation, even whales and snakes, because there are fossils of both which show external limbs for them as well, and at some point, both had toes. This is where we realize that defining life by their characters alone is insufficient, because those may vary. After all, a human born severely retarded, or without arms, -is still a human, even if they don't match all the criteria we normally cite for our species. So the only way to consistently define living things accurately is according to their ancestry, and not just by their current physical characteristics.
So to recap:

Stegocephalians are a subgroup of Sarcopterygiians; organic RNA/DNA protein-based, metabolic, metazoic, nucleic, diploid, bilateral, digestive, tryploblast, opisthokont, deuterostome coelemates with spinal chords, skulls, backbones, jaws, lungs, and adapted to four legs with toes.

Tetrapods are a sub-group of gill-less Stegocephalians with a skeletal structure modified to operate their four limbs effectively. This group also includes anything that is descended from quadrupedal stegocephalians, and that includes humans. 
One subset of Tetrapoda is Anthracosauria, pentadactyl post-aquatic 'terrestrial' tetrapods.
Tetrapods are divided into two sibling sub-groups: Anthracosaurs and Amphibia (modern amphibians). Some anthracosaurs were technically "amphibian" themselves, but they weren't like modern amphibians, which form their own clade.

One curiosity about modern amphibians are their lack of scales. I've heard creationist arguments that reptiles and fish both have scales but amphibians don't, and that fossil amphibians and their ilk can't be considered an intermediate group for that reason. But in reality, fish scales are made of dermal bone, and some extant amphibians, called caecilians still have them, (albeit sparsely) imbedded in their skin, which again is just what you would expect from an intermediate grade. "Reptiles" on the other hand have either epidermal nodes, (which look roughly similar to those on some newts) or they have follicular scales, which aren't present on any modern amphibian, and which are quite different from the scales of fish. The scales of a reptile are made of keratin and are sub-dermal, like caecilian scales, allowing the skin to be shed without losing a single scale in the process. Another keratinized follicle site is at the ends of the toes, the first claws, which eventually thickened into hooves on ungulates and fingernails on Old World monkeys.
One  one thing that was inherited from pre-amphibians are the configuration of the fingers. This where the still fish-like ichthyostega represents another important upcoming clade. At first, their ancestors had eight fingers emerging amid the rays of their fins. Then the front fins lost a digit, and the aft fins followed suit later on. But what happened next is interesting. What has now been reduced to a dew claw on dogs, a wing claw or leg spur on birds, and a thumb on you, -appears to have begun as a fusion of three digits into one, establishing the first pentadactyl configuration that you already know like the back of your hand.
Anthracosaurs are typically known by their pentadactyl extremities even though this trait is shared with Amphibia, their sister taxon. But Anthracosaurs also have skeletal features not shared by the other five-fingered fauna.

One subset of Anthracosauria is Amniota, dry-skinned digited tetrapods which developed in amniotic fluid,
and which have keratinized digits, (claws, fingernails, hooves).
"A third group of late Paleozoic tetrapods, the anthracosaurs, include the ancestors of amniotes. Like early amniotes, anthracosaur vertebral centra are dominated by the pleurocentrum. They also share similar phalangeal formulae (numbers of finger and toe bones) in the hand and foot. Large individuals were supported by two sacral vertebrae, [the basis of what would become a full pelvis] and some species had well-developed limbs adaptated to terrestrial locomotion. Some anthracosaurs were even thought to be amniotes until larval young were found to possess external gills (note: external gills, such as those seen in living amphibian larva, are not homologous with the internal gills of fish and Acanthostega). The amnion does not fossilize, so the key character for recognizing amniotes in the fossil record is the presence of two proximal tarsal bones (ankle bones next to the tibia and fibula, called the astragalus and calcaneum] [a complete ankle]. Anthracosaurs possess three proximal tarsals."
       --Department of Geology; University of Illinois at Urbana-Campaign
Solenodonsaurus janenschi is a transitional species between basal anthracosaurs and their apparently non-amphibious descendants. Known from a single, incomplete fossil, it shows loss of the lateral line on the head, which was present in amphibians, but still has the single sacral vertebra of the amphibian. Two other specimens known from the early Pennsylvanian period, (Hylonomus and Paleothyris) also show the sort of half-amphibian / half-reptile features which anti-evolutionists keep saying could not exist.
"These are protorothyrids, very early cotylosaurs (primitive reptiles). They were quite little, lizard-sized animals with amphibian-like skulls (amphibian pineal opening, dermal bone, etc.), shoulder, pelvis, & limbs, and intermediate teeth and vertebrae. [That's right, these were lizard-like except for their salamander heads and fish scales.] Rest of skeleton reptilian, with reptilian jaw muscle, no palatal fangs, and spool-shaped vertebral centra. Probably no eardrum yet. Many of these new "reptilian" features are also seen in little amphibians (which also sometimes have direct-developing eggs laid on land), so perhaps these features just came along with the small body size of the first reptiles."
           --Kathleen Hunt; Dept of Zoology, University of Washington

The emergence of amniotes.represents a significant evolutionary leap in that it is a developmental one, and such fundamental variance is especially rare. But this is owed to the equally-dramatic change of moving from aquatic nests to a wholly terrestrial existence.

The eggs of amphibians are dependant upon the water, but are laid along the shore to provide the most light and warmth (and because the parents can no longer breath water). Just as a peeled grape will develop a kind of skin when it is left to the air, so do the amphibian eggs, usually at the cost of the contents. After millions more years, some lucky generation had a kind of skin pre-installed that withstood accidental exposures to dry air. Obviously these survived to pass the benefit of that mutation on. After thousands of generations, subsequent offspring that had tougher and harder skins on their eggs also passed on those genes. Also in the eggs of this particular anthracosaurian sub-group, there were extra-embryonic membranes; the allantois, the chorion, an enlarged yolk sac, and an amnion filled with amniotic fluid. Organisms which develop in an amnion, are called amniotes.
"The comparative aridity of the terrestrial environment affects all aspects of amniote biology, and not just their reproductive systems. Thus, amniotes have a relatively impervious skin that reduces water loss. They also possess horny nails that, among other things, enable them to use their forelimbs to dig burrows into which they can retreat during the heat of the day. The imperative to reduce water loss is equally evident in the density of renal tubules in the metanephric kidney of amniotes, in the larger size of their water-resorbing large intestines, and in the full differentiation of the Harderian and lacrimal glands in the eye socket whose antibacterial secretions help to moisten and, along with a third eyelid (the nictitans), to further protect the eye from desiccation. The commitment of amniotes to a life on land is also revealed by an extensive system of muscle stretch receptors that enables finer coordination and greater agility during locomotion, their enlarged lungs (which are the only remaining organs of gas exchange owing to the loss of gills), and the complete loss of the lateral line system other vertebrates use to detect motion in water.
Many of these features are rarely preserved in fossils, but there are some novelties in the skeleton that are no less diagnostic of amniotes. For example, amniotes have at least two pairs of sacral ribs, instead of just one pair. They also have an astragalus bone in the ankle, instead of separate tibiale, intermedium, and proximal centrale bones. Finally, they have paired spinal accessory (11th) and hypoglossal (12th) cranial nerves incorporated into the skull, in addition to the ten pairs of cranial nerves present in amphibians."
                                            -- Drs Michel Laurin, (Université Paris) and Jacques A. Gauthier, Dept. of Geology at Yale University
Anthracosauria included some still very amphibian-like things as well as cotylosaurs, or "stem reptiles", which were the earliest amniotes known. These had a more-advanced brain which evolved from modules and paleocircuits of the amphibian brain, which itself is slightly more advanced than the archipallium fish brain.
This is a mammalian brain. All the colored bits of this image are present in reptiles, although some of the more advanced ones have a little bit of the gray cortex too. The amphibian portion of the brain is shown here in pink. As you can see, the "reptile brain" also known as the Limbic system or "R complex", ("R" for "reptile") can still be found in the core of your own brain.

"Comparative neurobiology suggests that the cerebellum first arises in fish, as a specialized structure of the lateral-line system (used for orientation of the organism). Primative fish, such as hagfish and lampreys lack cerebellar structures. However, bony fish (teleosts) have a well-definded cerebellum. Somewhere between these two types of fish, perhaps cartilaginous fish such as sharks, a structure anatomically and functionally similar to mammalian cerebellum first appears.
Fish have mostly archipallium, which may be a precursor to mammalian hippocampus; amphibians have archipallium and paleopallium and a group of basal nuclei, which are precursors to mammalian basal ganglia, such as the striatum. Neopallium first appears in advanced reptile brains, such as crocodiles. In mammalian brains, neopallium becomes greatly enlarged and is termed neocortex. The cerebrum in mammals is comprised of neocortex."
                         --N. Bradley Keele, PhD, Assistant Professor, Dept. Neuroscience, Baylor University
Not all modern amniotes still have five digits with fingernails on each extremity. Some, (like the three-toed sloth) have lost one or more of them. Artiodactyls (deer, cattle, giraffes, etc.) lost their thumb. And their index and pinky fingers have been reduced to nubs forming the "cloven hoof". The fingernails of all ungulates (perissodactyls and Artiodactyls) and Afrotherians (elephants, Sirenians, and aardvarks) have thickened into hooves. If you want to know what elephants and manatees used to look like millions of years ago, just look at a tapir. Perissodactyls (including the rhinoceros) lost their "thumb" and "pinky" toes. Of this group, horses lost all but the middle finger, where the fossil record shows they had more once upon a time. Cetaceans are the only group to lose their nails entirely, (which the fossil record shows were once hooved). However they still have all five fingers concealed in their flippers. Sea cows and manatees even have fingernails still visibly present on their flippers. Fossil snakes like Pachyrhachis problematicus, Haasiophis terrasanctus and Eupodophis descouensi reveal that snakes once had complete legs with five toes, each with claws on them, where only a few pythons still have any vestigial claws now, and they're only present on males. Some dinosaurs lost one or two of their fingers, which is why the birds which descended from them still don't have all five digits on their legs or arms (when they have them). But most of the earliest dinosaurs, (including some of the therapods) still had all five. The "thumb" toe on dogs has been reduced to a useless dew claw, and cats lost their dew claws altogether. Humans follow what is considered a "primitive" body plan in that they never lost any of these affectations on either set of their limbs.
So as a gill-less, organic RNA/DNA protein-based, metabolic, metazoic, nucleic, diploid, bilaterally-symmetrical, digestive, tryploblast, opisthokont, deuterostome coelemate with a spinal chord and 12 cranial nerves connecting to a limbic system in a jawed-skull attached to a vertebrate tetrapoidal skeleton with a sacral pelvis and ankle bones; and having lungs, tear ducts, and fingernails on all five digits on all four extremities, in addition to your embryonic development in amniotic fluid, humans meet all the necessary criteria to be an amniote.
One subset of Amniota is Synapsidae, amniotes with a single temporal fenestra.
"The first reptiles immediately split into two major lines which modified these traits in different ways. One line developed an aorta on the right side and strengthened the skull by swinging the quadrate bone down and forward, resulting in an enormous otic notch (and allowed the later development of good hearing without much further modification). This group further split into three major groups, easily recognizable by the number of holes or "fenestrae" in the side of the skull: the anapsids (no fenestrae), which produced the turtles; the diapsids (two fenestrae), which produced the dinosaurs and birds; and an offshoot group, the eurapsids (two fenestrae fused into one), which produced the ichthyosaurs.
"The other major line of reptiles developed an aorta on left side only, and strengthened the skull by moving the quadrate bone up and back, obliterating the otic notch (making involvement of the jaw essential in the later development of good hearing). They developed a single fenestra per side. This group was the synapsid reptiles. They took a radically different path than the other reptiles, involving homeothermy, a larger brain, better hearing and more efficient teeth. One group of synapsids called the "therapsids" took these changes particularly far, and apparently produced the mammals."
                                               --Kathleen Hunt, Zoology dept. University of Washington
The first amniotes were captorhinids, which are considered the first "reptiles". This is a marvelous example of the much-debated punctuated equilibrium because the dry land represented a totally new set of environments to be colonized; And among vertebrates, only amniotes had the capability of doing that. So there was an explosion of biodiversity very quickly.
So right away, we have a division into anapsids, diapsids, and eurapsids on the one side, and synapsids on the other. And each of these led to a whole tree of fossil species.  Anapsids (no temporal fenestra) lead to several interesting forms, of which only turtles still survive. But once upon a time, there were turtles on the half-shell, with partially-formed shells, or with no shell. There were also some fairly bizarre forms that weren't so turtle-like.

Diapsids (two temporal fenestra) were the "true" reptiles, class; Reptilia. They divided into Archosaurs on one side, (which eventually lead to crocodilians, phytosaurs, pterosaurs, dinosaurs, and birds) and Lepidosaurs on the other, leading to sphenodons, squamates (lizards and snakes) and numerous other varieties which are now all extinct. An interesting side-note about this division is that archosaurs all have a four-chambered heart where diapsids have a two-chambered heart, and modern anapsids have a sort-of three-and-a-half chambered heart. This in addition to their other comparitive morphologies and their stratigraphic position in the fossil record makes them another fine example of a transitional intermediary.
Eurapsids (two temporal fenestra fused into one) are an even more interesting story. They began as an offshoot of diapsids, and have several transitional species present to document their evolution into platyoposaurs, elasmosaurs, and icthyosaurs.
One subset of Synapsida is Therapsida, synapsids with mammalian skeletal formations.
These transitions occurred during the Carboniferous and Permian periods, the late Palaeozoic era, which is my favorite epoch in geologic history. How I wish I could go camping or fishing in this period, and experience the wilderness of this world with my own eyes! To see a landscape so pristine, and just different enough from the modern world to still seem alien. At this time, there were as yet no crocodilians, but there were phytosaurs which were just like them, except that they were much bigger, and were sort-of sabre-toothed. There were also a collection of sort-of half-crocodile / half-dinosaur transitional species wandering about. But the most interesting group were the pelycosaurs, a paraphyletic troup of synapsid "reptiles" which began to display very mammal-like traits. From this group came the sphenocodonts. These lead to therapsids, which demonstrated a series of transitions in jaw structure, musculature, skeletal adaptations, and the configuration of the inner ear bones, all of which are unique to mammals. For those seeking part-reptile / part-mammal transitional species, therapsids represent the best-documented transition between vertebrate classes available in the fossil record.
Synapsids have only one temporal fenestra, which on humans is now the auditory meatus, or the "ear hole". Early synapsids were quite reptilian, and included pelycosaurs like the dimetrodon, (pictured below). Synapsids soon branched into increasingly mammal-like therapsids.
"While living mammals are well separated from other groups of animals today, the fossil record clearly shows their origin from a reptilian stock and permits one to trace the origin and radiation of mammals in considerable detail."
    --Phillip Gingerich, professor of Geological Sciences, University of Michigan
One subset of Therapsida is Cynodonta, therapsids with canine teeth.
Procyonosuchus by Richard Hammond
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Of course all throughout this period, there were dozens of creatures unlike anything you would recognize today.  Much of this development came before the first dinosaurs ever appeared. The earliest known synapsid reptile, Protoclepsydrops haplous (early Pennsylvanian) still had amphibian-type vertebrae with tiny neural processes. Evidently, reptiles had only just recently separated from amphibians. Archaeothyris (early-mid Pennsylvanian) still had primitive amphibian/early reptile-like features in the jaw. But it also began to show a slight hint of the different tooth types typical of mammals. Haptodus (late Pennsylvanian) -- One of the first known sphenacodonts, had several skeletal features becoming more mammalian, particularly in the teeth, which began to show the first true rooted canines, and not the sort of fangs snakes have. Subsequent species lost the last vestiges of strictly-reptilian bones, and developed the ear drum, another exclusively mammalian trait. Throughout this sequence, we also see an improvement in the ligaments and muscularity to show a steady progression from very primitive lizard-like things to more advanced and adaptive "reptiles" that were also arguably mammals of one sort or another at the same time. In fact, there were several of these which blur the line between reptiles and mammals so much that in some cases, its difficult to state which class these things should belong to. Procynosuchus (latest Permian) the first cynodont, was already a sort of dog-like pseudo-lizard which quickly begat some very lizard-like primitive quasi-mammals, like thrinaxodon. These early Triassic cynodonts had very definite canine teeth and are considered by many to be one of the first mammals, even though they weren't quite complete mammals, and still bore some vaguely-reptilian vestigial traits. These were also among the very few mammal-like semi-reptiles to survive the Permian extinction, an event even more devastating than that which later brought on the demise of the dinosaurs. By the time we get to things like Cynognathus (early Triassic, but suspected to have existed even earlier) we have a nearly complete mammal with just the slightest reptilian traits, like the as-yet undistinguished uniform reptilian-style cheek teeth behind the definitely mammalian canines.
And of course all of these show a progressive increase in the neocortex of the brain. So these creatures were not only much more intelligent, but also capable of emotional feelings, even some degree compassion, a trait that is mandatory given the mother's need to nurture and defend her young beyond what any mere reptile would ever do. These forms appeared at about the same time as the first dinosaurs by the early Mesozoic era. After that point, it seems there was little substantial development among mammals until the dominant dinosaurs were finally eliminated at the end of that era, some 150 million years later. Until then, every group of them, including the relatively large (medium dog-sized) dino-eating Repenomamus, all bore a striking superficial resemblance to either shrews, rats, or opossums, which all look alike anyway. These are considered the most "primitive" of mammalian shapes because of that, and for the fact that their tails still have real scales in addition to hair, which (under a microscope) is only a modified scale.
One subset of Cynodontia is Theria, (mammals) endothermic (warm-blooded) therapsids with lactal glands.
an almost opossum-like dinocephalian cynodont
(who's tail was probably naked and scaled).
Modern mammals are defined as warm-blooded, fur-bearing animals with mammaries. It has so far been impossible to determine when any of these traits first appeared since there's little chance of any of these traits leaving any fossil traces. But there are obviously many other details which are particular to mammals alone, and these indicate that there were once at least two or three more distinct mammalian sub-classes than there are now co-existing with the last dinosaurs, and some of which died out with them. All of the millions of mammals you can think of which exist now are members of only three surviving sub-classes; monotremes, and the marsupials & eutherians which both appear to have flowered out of sister stock.
Cynodontia          ___Triconodonts
Diviniidae         /_______Monotremata
Theria (mammals)__/__Multituberculata
Procynosuchidae   \   _____Marsupialia
Galesauridae       \_/_Palaeoryctoids
Thrinaxodontidae     \________Eutheria
The greyed-out names in the paraphyletic column above represent extinct reptile-like quasi-mammalian forms unlike anything still living today, and from which true mammals emerged.  The blue names in the cladogram represent the three groups which account for every mammal species to survive beyond the dinosaur age.  Every mammal people have ever seen belongs to one of these three groups.  Everything else in this list were all extinct millions of years before any human era.
One subset of Theria is Eutheria, mammals which are born in a placenta, and which have nipples.
Similarly, if we didn't have any extant marsupials, (kangaroos, koalas, opossums, etc.) then we could never have guessed about the bizarre methods of birth and nurturing they use. So there's just no way to know how extinct classes birthed their young. As for lactation and endothermy, we know that some Triassic proto-mammals were burrowing animals, and this indicates that either could already have been factors, since they nested with their young, apparently beyond the hatchling stage. And since this trait is shared by all existing mammal groups, it would have to have been inherited from a common ancestor of all them, and that would have been one from the Triassic, since the first eutherians, (placental mammals) seem to have appeared by the mid-Jurassic.

One hypothesis for the evolution of lactation is that the disaccharide, lactose evolved from a sort of post-natal "sweating" of lysozyme, an ancient protein widely-distributed in the animal kingdom. As the young cling to the mother, often with their little semi-reptilian mouths, they would naturally ingest some of her sweat, and the development of actual lactose synthase glands would have a preferential selective pressure as a result.

So, whether you accept the evident evolutionary origin of all of your mammalian traits or not, do you at least accept that as an amniote with a single temporal fenestra, that you should be classified as a synapsid?  If so, then as a synapsid with a dramatically-enlarged cerebreal cortex in addition to each of the modifications listed in the illustration below, do you accept that you should also be classified as a therapsid?

There's no way to know how any of these extinct sub-classes reproduced. It would be reasonable to assume that (except for paleorectoids) they likely laid eggs like modern monotremes, but there's no way to be sure if they did. If we didn't have any extant monotremes, we wouldn't even have been able to see how mammals made the move from eggs to live birth. The leathery "shells" of platypus eggs are so thin and membranous that they're hardly shells at all, and they hatch almost immediately after they've been laid. So its almost a live birth, almost a placenta rather than a shell. And that's what distinguishes Eutherians like yourself from other mammals.
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