Despite many legends across virtually every culture, Thorndyke and Stevens, the two most reliable sources on the topic, identify only two types of dragons: the Elder and the Younger.
According to Stevens, the Elder dragons are the massive, semi-reptilian leviathans of legend. They are virtually immortal, hundred or more yard long, practically indestructible creatures. They are said to possess incredible senses, speak all known languages, and be highly resistant to all forms of magic. Stevens notes that Elder dragons also exert significant control over fire and air. Fortunately, they appear to be very few in number. The dragons will not discuss the subject, so it is difficult to tell exactly, but Stevens estimates that there may be as many as twenty Elder dragons around the world, certainly no more.
Much more common are the Younger dragons, studied extensively by Thorndyke. These dragons appear to be human, or are able to assume a human-like appearance. But, Thorndyke notes, they can choose to manifest draconic traits such as claws, scales, fangs, even wings and tails. They also appear to be resistant to magics and impervious to heat and fire. Although not on par with their elders, Younger dragons possess exceptional strength, senses, and linguistic talent.
Younger dragons are relatively common around the world and regularly interact with both humans and magic wielders. Usually these relations are on behalf of Elders, but some are on their own account. Most dragons, Elder and Younger, seem to mostly work against each other. Their complex interactions probably revolve around acquiring power and position within their society. Thorndyke attempted to make sense of the relationships, but eventually concluded that dragons have a tangled, arcane hierarchy.