Boss levels are levels that you must fight (or occasionally locate and fight) a boss. Almost all of these are either special "arenas" for each boss that can only be accessed by fighting that boss, or "dungeons" so to speak, areas of the game that are lengthy levels with a boss at the end. There are four types of boss level.
Arena- Storywise, these arenas can only be accessed once, have no level preceding them, and can't be left until the boss is defeated. They can be reentered in Boss Arena mode, however.
Single-Visit Dungeon- These lock you in them until the boss is defeated, and can't be reentered after the fight.
Limited-Visit Dungeon- These areas can be repeatedly left and reentered until the boss is defeated. Afterward, they cannot be reentered.
Unlimited-Visit Dungeon- A bit of a rarity, these areas can be accessed whenever you want, even after the boss is defeated. These have a tendency to be fairly lengthy, the kind of level where you have to locate and fight the boss. Often have collectibles hidden in them.