The Cat Fanciers Glossary defines purebred as, "purebred: A cat whose ancestors are all of the same breed, or whose ancestry includes crossbreeding that is allowed in the breed standard. For example, a purebred Bombay may also have Burmese cats in its background." Generally a cat's pedigree (list of ancestry) must be certified by the registry, before it can rightfully be called a "purebred."
A good example of breed confusion is the American Shorthair cat breed, which originally was called the "Domestic Shorthair" breed. Lacking that necessary pedigree, a domestic shorthair cat (DSH) can simply not be referred to as an American Shorthair cat. Learn more about How to Tell if Your Cat is a Purebred