Fatigue = How many rooms you can do of any dungeon per day. Every room in a dungeon takes away one fatigue point, once you are out of them, you can't enter a NEW dungeon, you can however finish the one you are doing.
Since EXP comes mainly from dungeon runs and quests (and quests depend on dungeon runs), once you run out of FP you won't be able to level anymore for that day.
In early levels fatigue tends to go fast since the dungeons are somewhat easier. Later on the entire fatigue bar will last around 4 - 6 hours or so.
It's highly recommended making at least 2 characters, you will surely get bored with just one.