Beat the game on normal or hard difficulty and obtain all of each character's chips to make their special armors available. For X and Zero, you simply need to obtain all of their chips (including the Sigma Blade for Zero) and defeat Lumine. After you beat Lumine and start a new game on the clear data, beat Noah's Park (the first level) and when you select your character, the seperate armors should be available.

X obtains the ultimate armor (not as cool looking as the ones from X4-X6) and Zero obtains his Black Armor.

Axl requires special circumstances to actually obtain his white armor, at least he did in my case. To get Axl's white armor, you must obtain all of his chips and deliver the final blow to Lumine with Axl using a double attack.

Axl becomes white and powder blue, and he actually looks a little cooler.