Here's a list of cat breeds from one of the largest breed registries in the world, the CFA. There are a number of organizations and governing bodies of the cat fancy worldwide, and the CFA is one of them.
Each has its own requirements and breed standards, thus making the lists from the various organizations different.
From time to time, new breeds will be recognized. Sometimes, it's a matter of recognizing a variant of an existing breed. Sometimes is just a matter of a coat color or length that was previously not allowed.
The Javanese, for example, is a colorful version of the Balinese. The CFA breed standard is the same for the two cats, except for the colors that are allowed for competition. Similarly, the Somali is essentially a longhaired Abyssinian.
Other times, breeders will create a new breed that is quite different from the ones it's created from. It takes time for the breed to develop, and even more time for the characteristics to be considered stable from generation to generation.
Then, an association has to accept the breed and subsequently, the breed is added to the list of breeds for that particular association.
The Cat Fanciers' Association (CFA) is the "World's Largest Registry of Pedigreed Cats" and, as of this update, recognizes 40 breeds in the Championship class, and 1 in the provisional class (The Ragamuffin).
None of the cats in the CFA registry are hybrid cat breeds, and there is a lot of controversy surrounding hybrids.
Update: As of 2016, the Bengal cat breed is now allowed to be shown, and is the only hybrid breed in the CFA.
This is a noticeable difference between the CFA and some other organizations in the Cat Fancy, such as The International Association of Cats (TICA). TICA has several hybrid cats in its ranks.
When choosing from among the different cat breeds, it may be helpful to review the breed profiles so you know more about what to expect from a particular breed (Persians need lots of grooming, Siamese are very talkative and active, etc.). Click the links below to read more about each type of cat.