Unlock the following modes by performing the corresponding tasks:

MODE - HOW TO UNLOCK
Stage mode - Beat a stage in Arcade mode on Normal difficulty to unlock it at the "Extra" menu. You can then replay that stage whenever desired.
Endless mode - Beat Arcade mode and all five stages in Stage mode to unlock Endless mode. In this mode, there are no Bosses, with the stages cycling endlessly.
Extra mode - Beat Arcade mode to unlock Extra mode after Arcade mode is selected again. In Extra mode, the difficulty level is increased.

ACHIEVEMENT - HOW TO UNLOCK
Sure Shot (10 points) - Defeat a wave of invaders without missing a shot.
Combo Meal (10 points) - Destroy at least 21 invaders in a row.
Goliath Killer (10 points) - Defeat the first Boss Invader.
First Taste of Victory (10 points) - Win an online match for the first time.
Lieutenant Rank A (10 points) - Achieve Rank A at the end of a level.
Lone Wolf Victory (15 points) - Beat the game in Arcade Mode.
Hot Streak (15 points) - Trigger 4 rounds of Fever Time.
Space Invader Veteran (20 points) - Won 25 online matches in VS Mode.
Team Victory (20 points) - Beat the game in Co-op Arcade mode.
Hero of Nagoya (25 points) - Achieve 1 ROW using the Nagoya Attack technique.
Space Commander Rank S (25 points) - Achieve Rank S 3 times.
Take That! (30 points) - Defeat a Boss Invader without missing a shot.