Until Generation V, when activated, the message "<Pokémon name> escaped using Run Away!" will appear. From Generation V onward, a banner with Run Away shows up instead.
Outside of battle
Run Away has no effect outside of battle.
Pokémon Mystery Dungeon series
In the Pokémon Mystery Dungeon series, a Pokémon with Run Away will become terrified and run from all Pokémon when its HP is down below ½. Leaders are unaffected. It also protects against being immobilized by moves like Mean Look, or Abilities like Shadow Tag.