How do you make a hybrid animal?

Depends what kind of Hybrid you mean, they need to be of the same species, two different kinds of horses, for example, can make an offspring, as well as two dogs, if two species are very closely related, such as the horse and the mule or the tiger and the lion, they can produce an offspring too.

You need to let them mate to produce this, anything more complicated then that requires a genetic recombinator, and those are pretty expensive.