I was surprised and amused by the amount of difficulty I had in breaking this topic down into categories that made sense. It started out as a short discussion of furry, but I realized I had to explain that, and then to explain the difference between a furry and a realistic animal av, and soon found myself accelerating backwards at great speed.
There's a distinction to be made between several overlapping kinds of not-human representations in Second Life. (In my taxonomy, nekos and vampires and others of that ilk are differently-human rather than not-human, and are a separate topic for another day.)
It is easy to tell the difference between a representation of a black-and-white alleycat and a representation of Sylvester, the would-be nemesis of Tweetiepie. It is rather harder to explain the difference between that representation of Sylvester and a representation of a gryphon: neither of them has a biological existence in RL, but Sylvester is a caricature of a well-known RL type whereas the image we have of the gryphon is a serious attempt to make sense of the descriptions handed down through explorers' tall tales. And what then to make of the distinction between an avatar based on the canonical cartoon images of Sylvester, and a home-made furry cat avatar? Is the attempt at realism (in replicating the well-known cartoon image) to be considered differently from the furry-av-builder's act of imaginative invention (which may result in a figure that would pass muster alongside Sylvester in a TV cartoon)? Very tricky.
Nonetheless, on we go. This needs to be posted before I go to work today, but I'll come back and polish it on the weekend. Comments, suggestions and counter-examples are welcome.
I want to suggest that there are two basic classes of avatars: realistic representations (human avs that look human, four-legged tigers, faithful recreations of mythological beings or characters from popular culture) on the one hand and abstracted, anthroporphized figures on the other. This is not to suggest any kind of valuation between the classes or their subtypes.
RL animal avatars are biologically realistic*: they are quads (or have two legs and wings), their body shapes and faces are realistic in shape and size, and they usually can perform the RL-appropriate sounds and animations.
Mythological animal avatars are mythologically realistic, as it were: fauns stand upright on cloven hooves and have little horns, werewolves howl bloodchillingly and can stand upright or run on all fours at will, gryphons are quads with extra wings, dragons have scales and enormously long necks, etc etc.
The distinction between these types and those that follow is in the avatar builder's intention: the first two attempt to reproduce faithfully a well-known image, whereas the next types are imaginative and abstract.
Furry avatars are not biologically or historical-mythologically realistic (though they may be pop-culturally realistic as in the case of Sylvester). They can be thought of as cartoons or as caricatures of animals: like cartoon figures, they stand upright and have mobile hands, their heads are set atop their necks and usually have human attributes (e.g. binocular vision, whereas most animals' eyes are set on opposite sides of their heads).
Tiny avatars can been seen as a sub-class of furries (though both communities would probably disagree) in that they are specifically very small cartoon or anthropomorphized animals.
Practically speaking however, the most significant distinction between these types is this: that there are flourishing self-organized communities of furries and tinies (and a community of dragons in Wyrms) but natural-humans and quad animals don't seem to organize in this way. Natural-human avs are so common and so close to RL that it hardly seems worth celebrating our shared biology: "Look, I too have five toes on each foot!"
* within the limits of the SL body mesh and prim and texture technologies, obviously.