Run Away is a Pokémon ability introduced in the third generation. It allows the Pokémon to run from battle freely.
Pokémon with this ability can always flee from battle regardless of trapping movies or abilities (including Ingrain). However, the ability has no effect on switiching. Run Away is not guaranteed to work in the Battle Pyramid.
When the ability activates, a message saying "<Pokémon name> escaped using Run Away!" will appear. From the fifth generation onward, just a banner for the ability with appear.
Pokémon Mystery Dungeon series
If a Pokémon with Run Away gets below 50%, they will become terrified and run away from all opposing Pokémon until its HP is over 50% again. It also protects against immobilization like from Mean Look.
Pokémon with Ability