How are specialized breeds of dogs, cats, horses, and other animals produced?

All the species you mentioned which can "crossbreed" are just that - single species which can breed within their species. All the different breeds of dog, for example, are the same species. Thus what they do is not "crossbreeding" at all. The exceptions that I can think of are: horses can breed with donkeys although they are different species, and also with zebras. This only happens in captivity, however - in wild situations, the different social behaviors of the species prevents them from getting together. And the hybrids produced are usually sterile.