Monster X actually consists of two phases, and you can only deal damage to his health during the second phase. In either phase though, you can only deal damage to Monster X when he moves away the armour concealing his weak point (pictured below).
When his weak point is exposed, you need to destroy the 4-yellow orbs at each corner of the weak point. Once you destroy those, you'll enter the second phase of the fight. You'll finally be able to damage Monster X by hitting him in his weak point when it's exposed, similar to the first phase. Only this time you'll actually damage his health too.
He's a very tough boss, so don't be surprised if it takes you awhile to beat him. I'd highly suggest using the level 3 blade since it makes quick work of the missiles he fires at you. The level 3 bubbler also works well.
Here's a video of someone beating the boss, which may prove helpful: