AMERICAN These dogs originally came to America from our British forefathers. After breeding them in America they became the American Pit Bull Terrier.
PIT refers to where the dogs fought (fight). The fighting pits were small and usually only contained the two owners/handlers (and maybe a "referee"). When necessary, owners would have to pull the dogs apart. If a dog redirected its aggression on a human and bit its handler, it was killed. By using this method, breeders and owners successfully bred out human-aggressiveness.
BULL refers to the early Bulldog from England. These dogs' early job was baiting bulls for the butcher. This later became sport, and when that was deemed "inhumane," owners took their dogs to the pits and turned the dogs on each other.
TERRIER refers to the Black-and-Tan Terrier and/or the English White Terrier with which the early Bulldog was crossed. As you may know, terriers are known for their high activity level, tenacity, etc. Owners who wanted their dogs to fight (and win) wanted to breed in this “feistiness.”