My time to shine!
Leo Rose is right, there's those main 4 breeds. However, the titular American Pit Bull terrier, because it isn't recognized by the AKC, is super variable in its standards- the UKC standard is nearly indistinguishable from the American Staffordshire terrier, and some register their dogs as both UKC APBTs and AKC American Staffordshires, with the main difference being the size range and commonness of the coat color liver/rednose/chocolate. There's also the American Dog Breeder's Association, which is mainly a kennel club just for the APBT with a few other terrier breeds less focused on, favors a more "gamebred" dog, smaller, more terrier-like, less stout/bulky in looks, with a working drive much stronger than the American Staffordshire terrier, also, by its ADBA standard, said to be less good with other animals*. Many breeders of the gamebred/working line APBT claim this is the only "true" or "real" pit bull (see image below). But, in reality, this concern isn't very practical; I'd say like 99% of the pit bull-ish dogs that just about anybody would call "a pit bull" that you see around are not these lean, wiry, working line "true" pit bulls, and most people who say they are breeding pit bulls, (which are a lot of backyard breeders, with a lot of those dogs being mixed breed). So, I say, "pit bulls" with space/uncapitalized or "pit bull type dogs" to refer to the 4 multiple breeds and mixes of those 4 breeds, but really it's just semantics.
*(Meanwhile, the AmStaff and Staffordshire Bull terrier, according to its AKC website is good with other dogs "with supervision", the middle rating that is also given to many other common breeds such as German Shepherds, border collies, and boxers, putting them as better with other dogs than other breeds like Rottweilers, Akitas, and Irish Terriers, which are "not recommended". The third rating being "yes, good with other dogs", which includes the Husky, Greyhound, Bichon Frise. American Bullies, on the other hand, by their kennel club standards, should be good with other dogs and even cats.)
The American Bully is recognized by the UKC, as well as some other organizations. A Classic Am Bully looks rather like a very heavy-set APBT or Am Staff, and are pretty nice looking dogs, IMO. However, the Exotic and Extreme type dogs.... well, suffice to say that I, and lot of others, don't care for them.
I agree there, too, about the extreme looking dogs...like english bulldogs, they will end up with a lot of health problems. The American bully is super variable in its standards, party due the different classes, some look very similar to AmStaffs, others are practically bulldogs, with some being as tall as a beagle and others nearly the size of a mastiff, and many in between. See second photo below of an "exotic" bully next to a classic American bully- both are called American Bullies by their breeders and owners, but they're rather different.
And furthermore, a lot the pit bull type dogs in animal shelters are mixes of one or more of those 4 breeds, with other breeds thrown in. I think it's pretty common for there to be dogs that are something like 3/4 Amstaff or APBT and 1/4 mutt, as pit bulls in some localities make up a large population of stray dogs that breed as they choose. In fact, the dog I own is probably something like this- I couldn't really tell you just by looking at her if she's a APBT mix or an Amstaff mix or possibly has a bit of American Bully or Staffy Bull Terrier in her. Say, here's the most recent DNA test results for pit bull type looking dogs I found by using #embarkvet, the only test that tests for all 4 of the breeds. It can be hard to guess a mixed breed dog's heritage just by looking! And a lot of mutts seem to have a high amount of pit-ish breeds, and a lot of pits or APBT/AmBully/AmStaff crosses have a dash of something else. Many say these tests aren't always accurate (this is the most brand, as in the one that tests the most markers) but that just goes to show how complicated DNA and breed is. But DNA is DNA, these dogs do share about that percentage of DNA with the other dogs in the database that make up the reference for that breed, but we don't always quite understand what that means- many genes interact to control one trait, purebred dogs aren't copies of one another and instead share a lot of DNA (as much as human second cousins), and someone making this test must figure out what genes are shared by a breed, and what dogs in reference, prior to the test, were considered purebred.