How are new breeds of animal produced?

by having a male of one species and a female of another species mate.

for example, a liger is a cross between a male lion and a female tiger.

a tigon is a cross between a male tiger and a female lion.

a mule is a cross between a donkey and a horse.

sadly, none of these can create a new generation, because of the number of chromosomes they have and so on.