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.
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.