It should be "cat names".

It depends on the context. "Cats names" is not correct in any context. "Cats' names" can be correct: "My cats' names are 'Fluffy' and 'Snowball.'" Here, we are using the possessive form of the plural. My cats have these name. We could also say "Fluffy and Snowball are common cats' names." In "Cat names" "cat" is being used as an adjective, which is fairly common in English and becoming more and more common all the time. I live in a state where my driver's license says "Driver's License" on it. But in many states, they say "Driver License." In this case, it means "this is a kind of name that is used for cats, rather than for dogs or humans." We could say "'Fluffy' and 'Snowball' are common cat names." But we could NOT say "My cat names are 'Fluffy' and 'Snowball.'"