Pokemon can have anywhere between one and three Abilities. Two Abilities will appear randomly most commonly. For example, Pelipper's first two Abilities are Keen Eye and Drizzle. The third Ability, if unique, is Hidden . This means you won't see it if you breed or use Ability Capsule.
Breeding Keen Eye Pelipper will give lots of Wingull with Keen Eye and some with Hydration (Wingull's second Ability, Hydration, will become Drizzle upon evolution). But you'll never see Rain Dish , the Hidden Ability they possess.
Instead, you will have to SOS Chain Wingull or Pelipper until you have one with Rain Dish to breed. After that, you will have a high chance of hatching offspring with that Ability.
Passing Down Generic Abilities
In the realm of inheriting Abilities, the mother will always have the greatest amount of power. If you breed a female Pokémon with an Ability (the male parent can either be a compatible male or a Ditto, doesn't matter), the Ability she has will always have an 80% chance of being passed down.
So having a female Pokémon with the Ability you want maximizes the chance of getting said Ability. The male Pokémon's Ability has zero influence normally. However, if you breed a male (or genderless) Pokémon with a Ditto, they are suddenly treated as a female and pass down their Ability with a 80% chance .
Passing Down Hidden Abilities
Paired with a compatible male or with a Ditto, a female has a 6 0% chance of passing down her Hidden Ability, so it's slightly lower than with generic abilities. The other 40% will be split 20/20 if the Pokemon has two generic Abilities.
Breeding a male or genderless Pokémon with a Ditto also gives you a 60% chance of passing it down.
If you have a female Pokemon with the same species male Pokemon and both have their Hidden Ability, the chance for passing it down remains set at 60%, as this depends solely on the status of the mother.
Again, your friendly reminder that to get the base Pokemon with the Hidden Ability in the first place, you have to SOS Chain!