Nearly all domestic cats are not a breed of cat but a species of cat. The purebred cats, those cats which are selectively bred by cat breeders, are various breeds of cat of which there are about 104 maximum. Some of these breeds no longer exist or have failed to be successful. There are some more articles on this and associated topics at the base of the page.
The question is about the difference between a breed of cat and the species of cat. The word “breed” in relationship to cats refers to those animals which are bred selectively by cat breeders normally as a hobby in their homes. Each individual cat is registered with a cat association and it has a pedigree which shows the lineage i.e. their parentage going back about three generations. The whole situation is artificial because this is about artificial selection. The project is about making money and refining and changing the features of domestic cats in order to make one breed distinguishable from the other.
In contrast, categorising various types of cat into species is called taxonomy. The scientists have done this. It is an evolving process because of advancements in DNA analysis. There are about 36 different wild cat species. The concept of “species” normally applies to the wild cats. But the domestic cat is one of the 36 species.
So a domestic cat is a species of cat and within that umbrella category there are cat breeds as referred to above. Some people say that there are more than 36 species of cat and of course I accept that because the experts have not decided yet how many there are. Although the matter is quite settled but the project isn’t finalised.
As nearly all domestic cats are random bred and therefore not purebred, pedigree cats, you can regard them as a species of cat as I’ve mentioned in the first sentence. It’s only the purebred variety which you can label as a “breed” of cat.
I hope that helps in understanding this rather fine difference which is quite difficult to explain. The bottom line though is that the concept of “cat breed” is entirely human made and artificial whereas the concept of “species” is a classification of the various animals of the world as created through eons of evolution by nature.
SOME MORE ON TAXONOMY: