Why breed Pokemon in the first place?

Breeding allows you to optimize various important factors of your Pokemon, mainly its ability, nature, IVs and egg moves. Usually you'd do this by giving one Pokemon of your wanted species together with a Ditto into daycare together, which will result in them producing an egg after some time that then hatches into the first stage of your Pokemon. Important to note is that, if neither of the two Pokemon in daycare are Ditto, the resulting egg will become the first stage of the female one.

Egg moves

Those are moves that your Pokemon can't learn by leveling up, use of a TM etc, but can learn if one of its “parents” in the daycare already knows it. Then it will hatch and already know the move. A list of these moves can be found on various other sites and would exceed the scope of this guide by far (you can find some sites under ressources). These also list possible parents, as not all Pokemon can breed with all other Pokemon. If both Pokemon are from the same evolution line and also have the same Pokedex number, they will “teach” egg moves they already know to each other.

Nature & Ability

Usually the hatched Pokemon has a random nature. However, if one of the two parents holds an Everstone, then the hatched Pokemon will have always the same nature as the parent with the Everstone. The hatched Pokemon will have the mother's (or father's, if the other one is a Ditto) ability in 80% of cases, and the other basic ability in 20% of cases. If the parent has a hidden ability, that one is passed down 60% of the time, with each basic ability making up 20% respectively.

IVs (Individual Values)

IVs are values between 0-31 that are like “genetic predisposition” so to speak. They influence a Pokemon's stats and are set either when hatched or when you encounter the wild Pokemon.

IVs are a little more difficult to optimize. Usually, 3 are taken from the parents (they can come from both parents, in any combination) and the other 3 are randomized entirely. If one of the parents carries the item Destiny Knot, instead that one passes down 5 of its IVs and the 6th one of the child is set randomly. There are also items to choose one specific stat to pass down. However only the ones for attack and speed IVs are really relevant here for specific strategies, as usually you'd want all 6 stats to have 31 IVs optimally. It should be noted that AT LEAST one stat is always randomly generated.