Ok, I'm far from an expert and I'm greatly simplfying things to laymans terms, so true aeronautical engineers please correct me.
CLmax as was said is the maximum lift coefficent. Typically the higher the CLmax, the more lift your wing produces (per area) and the the higher the maximum angle of attack (AoA) you can pull in a manuever (lets call it a turn). I *think* its more applicable to sustained manuevers, not instantaneous.
So generally, if two planes have the same sized wing and the same wingloading, the one with the higher CLmax would be more manueverable. But nothing is without cost.
In general, to get a wing with a higher CLmax, you must have a thicker wing. Which means you have more drag, and a lower speed (all other things held equal).
This is all very simplified, but you get the idea.