Boss mode:
Beat the game to unlock Boss Mode at the mini-game menu. You'll fight all the bosses with only 1 life.

Extra mode:
Beat Adventure mode with 100% to unlock Extra mode at the Adventure mode menu. Extra is just a much harder mode.

Meta-Knight mode:
Beat Extra mode with 100% to unlock Meta-Knight at the mini-game menu. You'll play as Meta-Knight.

Thanks to Revolution readers Timothy, Jc Wilkinson, and Kirbymario!