A pure breed of cat is one that specifies the lineage of the cat, and means that it has no other breed mix in its blood or heritage than that which it is named.
Many people who adopt or purchase a cat do not think in the terms of "purebred." They fall in love with the cat or kitten's characteristics, personality, and behaviors.
There are some, however, who have a particular breed of cat in mind when purchasing or adopting a cat. Maybe they fell in love with the look of a particular breed, or they need their cat to be hairless, such in the case of the hairless cat breed. Or perhaps they just like the status of saying they own a purebred.
When you get a purebred cat, if purchased from a reputable cat breeder, you are getting a cat that has a long lineage of cats from the same breed. Every time a litter of kittens is born, each kitten is registered with the Cat Fancier's Association (CFA), and they can trace each kitten's heritage with registration papers to see its long history of pure breeding.
Purebred cats are mated with their like breed; for instance, a purebred Persian with another purebred Persian, and Siamese with a Siamese, and so on. There cannot be any other breed of cat in the lineage or the offspring will not be considered purebred.
One big misconception that's often made when speaking about cat breeds is with calicos. Many people think calicos are a type of breed when they are not. Calico refers to the coloring of a cat, and not a breed.