If a Pokémon holding an Adrenaline Orb is affected by Intimidate, the item will be consumed and increase the holder's Speed by one stage.
Outside of battle
From Pokémon Emerald onwards, if a Pokémon with Intimidate is in the first place in the party (even if fainted), there is a 50% chance that a Pokémon that would be encountered will not appear if its level would be 5 levels or more below the leading Pokémon's. It shares this effect with Keen Eye.
Pokémon Mystery Dungeon series
In the Mystery Dungeon series, Pokémon attacking a Pokémon with this Ability with a physical attack will have a 20% attack penalty.