MP is short for mana/magic points, and it's the amount of energy a player has at their disposal. All class skills need MP in order to use them. If MP hits 0, you will be unable to cast magic or use skills.
Ways of healing MP are to level up, drink a MP potion, leaving a dungeon (will be back at 100% MP), or to wait for MP to regenerate naturally.
Ways of increasing Max MP are wearing equipment or crests that increase MP or Intelligence, leveling up, and learning/leveling the skillMental Fortitude.
Increasing MP Regen rate can be done through equipment, crests, set effects, and the skill Attuned Mind.