In a nutshell, these miniature dragons, described as resembling infant Red Dragons in all ways bar behavior, are iconic for their ability to bond with wizards and sorcerers as familiars. Why? Rule of cool: after all, what could possibly be more badass than playing an arcanist with a miniature dragon riding around on their shoulder?
Personality-wise, the pseudodragon has remained quite consistent throughout the ages; playful, fickle, and demanding attention, they are usually likened to cats. Just cats that can fly. And have telepathy. And breathe fire. This makes rather demanding familiars, but sheer cool will usually see their masters striving to keep them satisfied and happy.
Pseudodragons first appeared in the original Monster Manual for D&D. They then reappeared in the Monstrous Manual for Advanced Dungeons & Dragons 2nd edition, where they gained the "Dragonet" typing, before appearing in the Monster Manual 1 for Dungeons & Dragons 3rd edition.
They were mysteriously absent in Dungeons & Dragons 4th edition, although the "Dragonling" familiar presented in Arcane Power may have been intended to be a more "generic" version of the pseudodragon, one where the player would have greater official control over its appearance and personality.