Dragons fall under a set of lineages or Classifications. They are classified based on their physiology. The current classifications are the following:
Caecerta (Reptile-like)
Mollcerta (Mammal-like)
Nullvolaris (Wyverns)
Serpensa (Serpent-like)
Drakes (Wingless dragons)
Caecerta (Reptile-Like) and Mollcerta (Mammal-like) are you stereotypical western style dragons. Typical quadrupedal, four limbs and a set of wings.
Nullvolaris are your Wyverns. Typically bat-like in appearance where their front limbs are their wings.
Serpensa are serpent, or snake-like dragons. These dragons have long bodies and can vary in their wing and limb numbers.
Drakes are wingless dragons, typically quadrupedal with four limbs.
There are non-draconic lineages as well. These are less studied, but still have classifications for them as well. The current non-draconic classifications are the following:
Griffin
Hippocampus
Hippogryph
Kirin
Griffins are a mammal-like species that resemble a combination of a cat and a bird.
Hippocampus are aquatic, horse-like creatures.
Hippogryphs are a mammal-like species that resemble a combination of a horse and a bird.
Kirin are scaled, horse-like creatures, that are almost dragon-like in appearance.