What animals can mate with different breeds?

Yes.

Sometimes this is intentionally set up by humans. Mules, for example, are produced by a cross-species mating between horses and donkeys.

Some happen naturally; many species of felids can interbreed if they're close enough in size, and it's fairly common to find crossbreeds of whatever the native small wildcat is with domestic cats. In some places, such as Scotland, this happens so extensively that the true Scottish Wildcat is now quite rare. Even larger felids can interbreed to produce animals such as the liger or tigon (male lion-female tiger and male tiger-female lion, respectively).

In many cases, the crossbreeds are infertile (mules, for example, are almost always sterile). However, this is not always true.