Polydactyl cats are feline X-Men.
Their mutant powers (as observed by the DejaMorgana Feline Observation Team, or DeMFOT) are Super Clumsiness, Picking Up Things That No Normal Cat Would Bother With, and occasionally Opening Doors. Polydactyl cats are the antithesis of the standard feline grace. Mine steps on her own paws with almost every step she takes.
Seriously, polydactylism is a trait found in cats of almost all breeds. An estimated 1 per 1000 cats are born with extra toes. While normal cats have five toes on each front paw and four on each back paw, polydactylism gives them up to seven on the front and, usually, five in back. The extra toes can grow on just one foot, front paws only, or all four paws. I've never heard of a cat with extras on the hind feet only, but they may exist.
There are no breeds that are known to produce more polys than average, but some breeders have been able to get higher poly birth rates by mating polys to polys. However, this practice seems to enhance other undesirable defects, and has in one infamous case resulted in a variety of cats with short, twisted front legs, back problems and extreme polydactylism, bringing the wrath of outraged animal lovers down on the breeder via the Jerry Springer Show and a massive hatemail campaign.
The extra toes do look a little bizarre on first sight. They make the feet look much bigger, and a good many of these cats end up with the creative names "Bigfoot" or "Mittens". In many cases, one or both of the extra front toes are set at an angle to the rest of the toes, so that they look like thumbs. They are, of course, not opposable, but they do give the cat a little extra grip on small things, and sometimes doorknobs.
As anyone who has ever trimmed a cat's claws may imagine, polys do take some extra care, and I wouldn't recommend them to people who like low-maintenance pets. In most cases, the extra digits are not a health hazard. However, sometimes the extra claws grow abnormally fast. My poly's extra claws are all normal except for one, which grows frighteningly fast, extra thick and sharply curved. If I didn't trim this claw every couple of weeks, it would probably grow into the pad of that toe and do some damage. As this cat is also extra protective about her toes, this can be a real pain. Declawing is not recommended for these or any other cats.
I would be remiss in noding about polys without mentioning Ernest Hemingway. The man was infatuated with polydactyls and strongly influenced by the popular superstition that they brought good luck. Estimates vary, but the Hemingway Museum's website claims that he had 50 polys. Many of their descendants still live in the museum in his old home in Key West. Many Americans call polydactyls "Hemingway Cats".