|Purebred Burmese cat|
Photo by Ma1974
A purebred cat is a cat of a recognized breed. These are cats that have parents of the same breed and so on for several generations. When the parentage is recorded by a cat association the cat is also a pedigree cat. The two go together in practice. Although you can have purebred cats that are unrecognized by cat associations such as the Bahraini Dilmun. This cat is arguably a purebred cat.
Purebred cats are all about ensuring that the lineage or parentage is pure. This means avoiding cats that are genetically not of the same breed.
The trouble is that when you do this to the letter you are likely to end up with genetically homozygous individuals. Or individual cats that are inbred. This is why the laws of marriage (for humans) in the UK prevents marriage to close relations. This rule is in place for health reasons as inbreeding results in health issues such as congenital defects, infertility and suppressed immune systems. The Burmese cat has a number of genetic illnesses.
Wild cats in captivity are frequently not purebred despite giving an impression to the contrary. The Bengal tiger in captivity will often be a "generic tiger" or cross breed but not of pure genes. Other wild cats, the small wildcats in captivity, will have little chance of survival because their population size is so low that they are inbred resulting in an unsustainable population.
Breeders of purebred cat sometimes have to outcross to other breeds and moggies to improve the genetic diversity. There is a constant battle between health and appearance. Appearance usually wins.