A lot of different questions from a new player...

[Enkidoh]

That would be the Limit Break gauge - it's basically a special ability anyone in the party can use - the exact effect of which is dependent on what class role the person who uses it currently is (a damage dealing class does a powerful physical attack, a healer heals the party, a tank buffs the defense of everyone, and mages do a powerful magic attack).

Naturally the more filled the gauge is when it's executed, the more potent the effect, hence most players like to 'save' the Limit Break for the end of the fight to 'wrap it all up', and usually will prefer a mage (like BLM or SMN) to use it only, as the attack can be extremely powerful (at full strength the mage LB casts Meteor that does enormous damage to all enemies). That's basically it in a nutshell.