Natural Cure heals any status conditions upon switching out or completing a battle (if it was in battle at the end of the battle).
This effect will occur if Natural Cure is the Pokémon's Ability in battle when it switches out and is not suppressed; a Pokémon that obtained the Ability with Trace will have its status condition cured when it switches out, whereas a Pokémon which has Natural Cure but has been affected by Gastro Acid will not.
Outside of battle
From Generation IV onward, Natural Cure automatically heals status conditions upon the end of a battle.
Sometimes it cures every single bad status condition the Pokémon has!
In the anime
In The Power of Us, during two simultaneous disasters in Fula City, Toren and his fellow Scientists attempted to create an elixir out of Natural Cure in the Pokémon Research Pavilion. However, a blackout occurred before the solution could be finished, forcing Toren to improvise with Team Rocket's Lum Berries.
In the manga
In the Pokémon Adventures manga
The Pokémon is healed from its status conditions upon switching out.
Due to the user interface displaying how many of the opponent's Pokémon have status conditions, even though Natural Cure does not have an activation message, the user interface will reveal that a Pokémon had Natural Cure the next time the Poké Balls representing the opponent's party are displayed.
In Generation III, this is the next time the opponent switches Pokémon.
In Generation V onward, it is visible on the move selection screen.