For a dog to be registered with the A.K.C., the dog's parents must be registered with the A.K.C. as the same breed, and the litter in which the dog is born must be registered with the A.K.C. Once these criteria are met, the dog can be registered as purebred by the A.K.C.
The registration only specifies that the dog is purely of one recognized breed -- it does not specify that the dog comes from healthy or show-quality blood lines. Neither is it a reflection on the quality of the breeder or how the puppy was raised.
Registration is only necessary for breeders (so they can sell registered puppies) or for purebred dog show participation (similar to the medieval requirement of royalty for jousting competitions).
A.K.C. is not the only registy of purebred dogs, but it is the one most Americans are familiar with.