There appear to be a number of theories as to why the slang for the vagina (or perhaps more precisely, the vulva) is pussy. The most common derivation that I see in dictionaries and other resources is that it can be traced back to the Old Norse puss meaning "pouch" or "pocket". In that sense, it is not unlike the etymology of vagina which itself comes from the Latin vagina meaning "sheath" or "scabbard".
Somebody asked me earlier today why a penis is also called a cock. I assumed that it had something to do with the rooster (the male chicken) being called a cock. While this is true, there's a little bit more to tell. Here's a brief investigation into the matter.
According to Etymonline:
Slang sense of "penis" is attested since 1610s (but cf. pillicock "penis," from c.1300);