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 Pokemon 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, Hidden Ability or no.
So having a female Pokemon with the Ability you want maximizes the chance of getting said Ability. Breeding a male Pokemon with Ditto won't give you the same chance.
Passing Down Hidden Abilities
If you want the maximum chance of passing down a Hidden Ability, you will need a female Pokemon with the Hidden Ability. Paired with a compatible male or with a Ditto, you have an 80% chance of passing it down.
Breeding a male or genderless Pokemon with a Ditto will result in a 20% chance of passing it down. In the case of genderless Pokemon, you'll just have to bite your lip and stay persistent!
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 80%, 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!