how can you identify cat breeds?

By looking at the papers from the breeder you got the cat from.

Unlike dogs, most cats aren't a breed or mix of breeds. Cats were domesticated thousands of years ago while breeds were only established in the last hundred or so years. And unlike dog breeders, cat breeders are very protective of their lines and tend to sell kittens altered. So most cats don't have ANY breeds whatsoever in their history - they're simply "domestics" - and make up over 97% of all cats.

To recognize breeds you can look at the pics on TICA's website http://www.tica.org/public/breeds.php Some breeds are quite distinctive - like it's hard to fake the flat face, round eyes and round ears of a Persian or the curled back ears of the American Curl. But lots of people make the mistake of thinking that every longhaired tabby cat they see is a "Maine Coon" or that their black cat is a "Bombay" simply because it's black.

Dogs can also be DNA tested to determine the breeds in their background, well, as long as the breeds in their background are common ones that they have already studied the DNA on. Meanwhile with cats no such tests to determine breed makeup exist nor will they likely ever exist. Cat DNA is nowhere near as diverse as dog DNA. Just compare the Saint Bernard to the Chihuahua. Clearly their DNA would have to be VERY different.

If you have a cat that you want to know the breed of just post a link to the pic and those like myself that know breeds can tell you what we think it is. But again, if it's not from a breeder and especially so if you've had it since it was a kitten and not from a breeder, then it's most likely simply a "domestic".

It is complicated. It is a combination of registration / certified pedigree and/or (conformation + proof of place of origin.)
I agree with most of what the others said about pedigree. , and that cats have less of a huge variation than dogs do, and that there are domestics who resemble breeds and a lot of people incorrectly think their non-breed cat is a breed.

but I am not going to say it is ONLY the papers because on the one hand there are the "natural breeds" and on the other hand there are cats with hung papers (dishonest pedigrees)

Most breeds could not be identified via a black and white photo. You can make a pretty likely guess if it's a breed with an extreme conformation developed via selective breeding, like the flat face of the modern Persian or the extremely angular, elongated look of the modern Siamese or Oriental, or as you say one of the breeds developed from a certain unusual mutation.

People who really know a breed can be pretty good at identifying that a cat is likely of that breed if they see the cat in person, but it helps if it's a breed that has something about it noticeably different from the typical moggies in the part of the world where they are.

There are 2 ways for a cat without a pedigree to get registered as a breed with a reputable registry without previously having papers from an established registry proving their parents' ancestry:
* One way is if the breed is currently classified as a "natural breed" -- that is a breed that first developed on its own in a limited geographic area, and then people have bred them because they thought it was something special worth preserving. In these breeds, a cat can be registered as that breed if it conforms to the standard and you have proof that it comes from the area of the breeds origin. Examples are Turkish Van , Norwegian Forest Cat, Korat

* Cat associations have a little-known policy in which a cat may be accepted into a breed if a experienced cat judges examine the cat in person and decide it conforms to the standard of that breed enough that it would be a valuable contribution to the breed. In TICA it is 3 judges , I think in CFA it may be 2.
This is not something they do lightly. They don't want it to be a way for someone to purchase a cat sold on a spay/neuter contract and without breeding rights, and then decide they want to go against the contract and breed it, and try to get judges to sign off on it and backdoor the cat into being able to register its kittens So you need more than the cat , you really need some proof about where that cat came from.
This is something that has been used in the same way as for the natural breeds for a breed that DID start as a Natural Breed but has been bred so long and its appearance changed, that it's not classified as an "Established Breed". The Persian and Siamese are examples. I know someone who was able to import Wichien-maat cats from breeders in Thailand (these are the cats the Siamese originally came from) and had 3 TICA judges examine them AND the documentation and ruled that they could be registered as Siamese. (as one judge said while examining one of the cats "there's nothing she COULD be but a Siamese."
But in a way they're not really saying it IS that breed... they are saying this cat meets the standard for this breed and may be a good addition to the gene pool. But by TICA's rules you can't show the progeny of such a cat until after 3 generations of breeding back to cats who DO have longer pedigrees.

Anyway so it's not like all the people who think their pointed kitten born to their black house cat is a Siamese or the grey stray cat they took in is a Russian Blue can just easily go get papers from legit registries saying they are those breeds. ( though in some cases it is done fraudulently by what is called "hanging papers". Unfortunately pedigrees for dogs and cats are based on the honor system. I would like to see more DNA testing proving the relationship to the sire and dam. (you can do that but they can't tell cat breed. though they've been able to see some regional differences in cat DNA.

You are right - there really is little difference among the breeds. Unless you have papers/pedigree or the cat is an excellant example of the breed, most likely the cat is a domestic long or short hair of no particular breed.

Many cats resemble a breed but probably don't even have any of that breed in them.

There is one way and only one way. They will have registration papers from their breeder. These papers will give all the information you need to know about the cat and what breed it is.

There is no other way to find out the breed of a cat. No tricks, no markings, no blood work or DNA testing.

You can identify color and patterns but that is not the same as identifying breeds. Pictures can not tell you what breed your cat is either. If it is not from a cattery who is registered with a cat association and has registration papers on the kitten/cat it is just a domestic long/short hair cat, that may look like a certain breed.

R P CAT

Certain eye shapes (oval or almond), face shapes (box, narrow or heart) can easily pinpoint a certain breed. Body structures too; some cats are far more elongated than others. I recommend "The Complete Cat Owners Manual" (i think its sponsored by Purina)... check it out from your local library. It has a fantastic section in the back full of colored photos of every recognized US breed... it was how i managed to figure out the my shelter kitty was actually a Balinese!