Comparing purebred and hybrid could apply to many species but I’ll use cats as an example.
Purebred cats have a pedigree which means that the cat’s father and mother, grandfather and grandmother, great grandfather and great grandmother and so on are recorded. And breeders have to breed according to the breed standard and meet the objectives of the breed standard which is prepared by a cat association. Breeders meet these standards by selective breeding. This means selecting the cats, the mating of which results in offspring that meet the aims of the breed standard.
Nearly always the selected cats are of the same breed hence ‘purebred’. When outcrossing is allowed it is to strengthen the genes of the cat because too much inbreeding causes a decline in health. Breeders get ‘hybrid vigour‘ with introducing a different breed.
Purebred cats are ‘pure’ in this sense. Unless specifically allowed by the cat association other cat breeds have not been introduced into the breeding program.
Cat breeds are one species of cat: the domestic cat.
There are 36 species of cat. All species of wild cat are naturally purebred and they are the planet’s wild cats. Sometimes humans contrive to mate one wild cat species with another. This can be successful: a lion with a tiger for instance. The resultant offspring are hybrids and are called ligers if the father is a lion and the mother a tiger. If the father is a tiger and the mother a lion the resultant cub is called a tigon. These matings only occur artificially and humans organise it. There are others (jaguar x lion).
To return to cat breeds. There are several types of hybrid cat in which a domestic cat has been mated with wild cat. The best example is the Bengal. This is a hybrid cat, specifically a wild cat hybrid being a cross between the Asian leopard cat and a domestic cat (for the F1 Bengal). Another good example is the Savannah, a cross between the serval and a domestic cat probably a Savannah cat. Please click on this link for a list of wild cat hybrids.
There are many examples of hybrid cat breeds (i.e. non-wild cat hybrids). For example the Bombay which is a mix of Sable Burmese and black American Shorthair. Please click on this link to see a full list with links to more. A classic hybrid cat breed has come to mind; the Maine Coon x Bengal. See two here.
Often we see on the internet cats for adoption which are not quite purebred (non-purebred but looking somewhat like a purebred cat). These are called e.g. Maine Coon mix. This means a purebred cat has been mated with a non-purebred (random bred) perhaps over two generations and the resulting offspring are non-standard Maine Coons. The same applies to all cat breeds.