In Generation V, female Pokémon with Hidden Abilities have a 60% chance of passing on the Hidden Ability to their offspring when bred with a male Pokémon from the same Egg Group. For example, if a female Darmanitan with Zen Mode breeds with a male Simisage, the offspring Darumaka may have Inner Focus instead of the usual Hustle, and would evolve into a Darmanitan with Zen Mode (instead of Sheer Force). Hidden Abilities cannot be passed down from male or genderless Pokémon, and will not be passed down if the female was paired with a Ditto.
In Generation VI, any Pokémon, including male and genderless Pokémon, can pass down their Hidden Ability when bred with Ditto. Ditto's Ability is irrelevent to inheritance.
Starting in Generation V, there is an 80% chance that the Ability slot of the female will be passed to the baby. For example, when breeding a female Venomoth with Shield Dust, the resultant Venonat has an 80% chance of having Compoundeyes rather than Tinted Lens. This chance does not apply if Ditto is used as either parent.