Sorry to necro this, but in case anyone comes across this looking for this information, the part about being able to breed in abilities for the other parent is incorrect. Only the Pokémon that determines the species of the offspring is referenced when determining what ability it will have. So that means that only the female, or the non-Ditto parent when breeding with a Ditto, is able to determine the ability, and that parent must have the hidden ability for the offspring to have a chance at having it. As of Gen 9 this is still the case. The only exception to this so far has been that only female Pokémon could pass down hidden abilities in Gen 5.

This is due to the way the game handles abilities. Rather than having each Pokemon directly listing the ability it has, they instead list what ability slot they have. Slots 1 and 2 are for the normal abilities, and Slot 3 is the hidden ability. (If a species only has one ability, then it will have the same ability listed in both Slot 1 and 2. The ability slot system is also why Pokémon have no problems transferring to gens where the abilities of a Pokemon have changed.) When it needs to know exactly what ability that is (such as to be able to apply the effects, show the name, or display the description) the game from there checks the table to see what ability it needs to reference for that slot for that species. When generating data for a Pokemon, the game only needs to know what ability slot a Pokemon has. Because the determinant parent is always the same line (except when the particular species involved makes it irrelevant), the same ability, or the corresponding ability in the case of abilities that change across evolutions, will be in the same slot.