In World of Warcraft, Normal means a couple of things:

1. A player's default condition. When a character spends time at an inn or in a city, the character's condition changes to rested. When the player uses up all of the bonus 'bubbles' acquired while resting, the player's condition changes back to normal. When in normal condition, the player's experience bar will be colored purple.
   See also Rest.
2. A type of Realm also called PvE.
3. Dungeon difficulty, there are two setting for post TBC dungeons, Normal and Heroic.