This is an overview of the species of dragon types that I will present in my Monster Manual. To give you a general idea, Drakes are primitive wingless and generally quadrapedal dragons of predatory animal intelligence. They are the size of a dog to the size of a horse, and rarely have breath weapons.
Wyverns are flying dragons that are avian in shape, having two legs and a pair of wings with no "arms". They always have a stinger of some sort. They rarely have a breath weapon and have rudimentary cunning intellect, but cannot talk.
Wyrms are serpent-like and have normal human intelligence range. They can speak but often don't and often have a breath weapon or some magical power. They tend to be brutish and evil.
Dragons (or true dragons) have 6 limbs, back legs, front leg/arms, and a pair of wings for the most part. They are exceptionally intelligent and always have a breath weapon and often know magic. they can be good and evil and are capable of complex planning and scheming. They, naturally, look at the other draconic species as inferior.
I steered away from the Chromatic/Metallic dragon flavors of D&D, I felt it was too formulaic, that some dragons could be known for their color but not all.