The concept of 'familiars' has existed in British folklore for many hundreds of years. Familiars are animals (some say animal-shaped spirits) that serve a witch in various ways, whether as servants, messengers or even spies. Historical accounts of witchcraft make mention of familiars; such animals have been credited with supernatural gifts, and even believed to be demons (or the devil himself) in disguise.
Familiars, in the strictest sense, do not exist within the world of Harry Potter. Although Hogwarts students are permitted to bring animals to school with them, the cats and rats we see there are, broadly speaking, pets. Ironically, the animal that acts most like a traditional familiar in the entire series is Mrs Norris, who belongs to the castle's only non-magical inhabitant, Argus Filch. It is true that owls are sent as messengers within the series, but this is in the context of a highly organised postal service, not unlike Muggle pigeon post.