In my world dragons are enviromorphic, meaning an individual's physical characteristics slowly change to suit its environment. They tend to keep certain traits, such as flight and some usage of their venom gland (not always to spit fire), but the range of dragon forms is quite diverse. Most of the dragons of the far tundras have replaced scales with fur, and those of the tropical climates usually have feathers, for instance.

So environment is one category, but just as important is a dragon's personality and behavior. Covetous "treasure-hoarding" dragons tend to be greenish and have long twisting horns, sadistic dragons tend to have evil looking sunken eyes, and so forth.

So if there was anything like an organized guild of dragon hunters (dragons are too rare and too hard to kill for that to be practical) they'd probably categorize them two-fold, by environment and behavior, but many dragons are a category unto themselves.

Also, what role do your dragons play in your world and how did they get there?

Dragons are immensely old, near-immortal, and don't reproduce. Many remember nothing of their (distant) homeland or origins, and those still hanging onto fading images or words can no more than speculate on their meaning. For most it's not a concern anyway. Dragons are generally content to the pursuit of a single pleasure or obsession. For some it's collecting (often precious metals and gems), for others it's the taste of mortal flesh, and for many it's simply sleeping without end (and upon being woken up by some interloper, exacting vengeance on whole continents).

Mortals generally go to great lengths to avoid them. It's understood that dragons keep dragons in line, since they're intensively territorial and those territories were drawn up thousands of years ago, so there's never been anything like an organized effort to destroy them. Dragon attacks are seen like natural disasters. Further, it's a component of many religions that dragons were created to be the protectors of this world. Not without some justification; a fear of dragons is a part of the reason that industrialization has barely inched past its beginning stages over the course of several centuries (the large scale extraction of natural resources is much harder), and why human civilization has kept largely to its old established territories.