The difficulty determines how easy or how hard the enemy you are fighting will be. For quests, you must start out at normal and work your way up to nightmare. Brawls, however, can be started on nightmare right after you unlocked them.
Normal is the easiest and initial difficulty level. Beating it unlocks Hard.
Hard is the middle difficulty level. It is unlocked by beating Normal and completing it unlocks Nightmare.
Nightmare is the most challenging difficulty level. It is unlocked by completing Hard.