Puzzle Quest, right? PQ had a significant influence on my game, though the core elements differ greatly: PQ uses puzzle mechanics in battle, and mine uses rhythm mechanics.
To answer your question (and give a quick summary of battle): each player, in tried-and-true RPG fashion, has Hit Points (shown at the bottom of the screen on either side). If your hit points hit zero, you lose, of course.
Each of the three subscreens represents a different part of battle. The red one is where the enemy attacks you; if you fail to hit these notes, you'll take damage. The glowing gems hurt you even more. If you hit them, though, no damage is taken. The green one is where you can cast spells, doing damage to your opponent, or setting up a barrier, or healing yourself, or various other effects. Casting a spell introduces a series of notes into this subscreen; you must hit ALL of them for the spell to take effect. Finally, the blue one is simply for regaining MP to cast spells.
Switching from subscreen to subscreen is indeed controlled by the player, using the L and R triggers. The game becomes not only about being able to hit those quick patterns of notes, but also strategically switching from subscreen to subscreen when necessary. All three need to be effectively used to defeat the opponent.
The game is rather complex; I'd arrogantly hazard to say it's the most difficult to understand of any of the DBP entries. Compounding the problem is the fact that it's a new style of game with no precedent or forebear, and consequently, it's almost impossible to get a handle on what's going on from the trailer. I wanted to spend time and make an easier-to-understand video, but I was hitting bug after bug after bug, and...sigh. Well, I'll do my best to hype it up come closer to release. = D And worry not, I have a deep (and optional!) tutorial system in place to ease the player in.
Thanks for the feedback! I was sad when no one responded. = (