Although there is one species of domestic cat there are between about 44 and 104 breeds of cat depending on your point of view, and hundreds of different kinds of cat coat colour and pattern. In addition, you will see, rarely, a mutated cat such as a cat who has become hairless because of a genetic mutation or has no tail (and other mutations). And there is a range of body types among the purebreds.
The answer to the questions depends what the phrase “different kinds” means to the questioner. I suspect that the question is about cat breeds and random red cats. The point here is that the number of purebred cats in the world compared to the overall population of domestic cats is really quite small. In fact the number of purebred cats compared to all domestic, feral and stray cats is going to be something in the order of perhaps 0.2%-0.5%. Therefore the fact that there are let’s say 70 odd cat breeds which are recognised, in the overall scheme of things it’s a rather minor statistic.
The biggest variation amongst cats is their coat. Of course the full spectrum of cat coat types is infinite. There mainstream coat types include solids (single colour), tortoiseshell, tabby, tipped, silver and pointed to which you can add white which almost doubles up their numbers. You might find about a hundred different mainstream coat types and if you add in white you might add an another 80.
You will also see quite a big range in body confirmation. What I mean is the body type of the cat because even among random bred cats you will see body type variation but nowhere near as extreme as among the selectively bred, purebred cats. You might interpret cats with slender bodies of a different kind to those with cobby (stocky) bodies.
Perhaps the best answer to the question is to say that there are two kinds of domestic cat, (1) purebred, pedigree cats and (2) random red cats. Both are the same species. The experts decide upon the species and how they categorise them. Categorising species is called taxonomy. The breeds are decided upon by members of the cat fancy i.e. the cat breeders. It is a pretty artificial construct because they decide whether to create and recognise a new breed or not. Whereas science dictates whether a cat is a distinct species or subspecies. Nowadays science relies upon DNA analysis in this respect. Taxonomy is still evolving.