Faint (ひんし/Near death) is the condition when a Pokémon's HP depletes completely (drops to 0). That Pokémon cannot battle anymore (however, it can still use in-field moves that it knows, such as Fly) unless a Revive, Max Revive, or Revival Herb is used, or it is healed by a Pokémon Center.
Sometimes during an adventure in gameplay, if there's a trainer fighting alongside the player character, they will heal the Pokémon after every battle or before a tough battle against a boss. Nurses and Doctors in Black, White, Black 2, and White 2 heal your Pokémon every time they are talked to after defeating them in battle the first time talking to them. Most of the time Pokémon will grow to like the player character more if they don't just rely on Pokémon Centers for healing, but using Revives and Potions to keep them from fainting, as long as a Pokémon is still healthy. A Sacred Ash can be used to heal all fainted Pokémon at once.
In the anime, fainted Pokémon have black swirls in their eyes.