Once you breed two monsters they will leave your party permanently. Before you rush into breeding you must understand how the breeding works. When two monsters create a egg their skills are passed on. If you mate a fire skilled monster with a lightning type monster you create a monster that can develop fire and lightning magic. A monster can only get skills that their parents have already learned.
Spells can also get combined.
Which can make a more powerful spell with two spells. Some of them need to be combined. Now another key is the pedigree of the monster. When you mate two monsters the first one is the pedigree. The monster will most likely be born as a type of the monster.