The best way to identify a cat's breed is to check its physical attributes against references online, or ask its veterinarian. Most house cats without pedigree papers are of mixed breeds, and it's easier to approximate what kind of breed they are most similar to.
Physical qualities such as length of coat, markings and colors, length of tail and body shape can help identify a cat's breed or at least the breeds it's composed of. Other details, such as character, playfulness and intelligence can help but are less precise, as cats tend to have very strong personalities regardless of breed.
As a general rule, unless a cat comes with pedigree paper it is not a purebred of "fancy" cat breeds such as Siamese or Persian. However, especially in the case of rescue cats, it's possible to find cats that resemble precise breeds more than others, such as a Balinese without a fluffy tail, or Siamese-mix cats with markings of different color than what purebred are picked for.
It's also important to consider what kind of area the cat was born in. It's generally safe to assume that a cat born in the United States is an American shorthair, as Asian breeds are usually imported and expensive.