In order to breed Pokemon, you first have to make it to the Pokemon Day Care Center on Route 117. Once there, talk to the woman at the counter, and she'll ask if she can take care of your Pokemon. You can leave two Pokemon with her at once, and provided the chemistry is right (and the Pokemon are a male/female pair), you may get an Egg the next time you walk through the Day Care doors. Make sure you have a spot in your inventory available to carry it. How do you know whether the two Pokemon you put in will yield a happy package Talk to the old man outside, and he'll give you a hint as to whether the union will be fruitful.
Breeding two of the same kind of Pokemon will either result in the same kind of Pokemon, or one of the previous evolution. However, if you breed two different kinds of Pokemon (they still need to be of the same Egg Type), you may end up with a completely different kind of Pokemon. As with trading, experiment with held items to produce new and different effects as well.