Question about Limit Breaks...

[Niwashi]

Anyone can activate the Limit Break. What it does depends on the class/job of the person who activated it. Melee DPS classes do a massive single target attack. Caster DPS classes have an AoE attack. Healers and ARC/BRD have an AoE heal for the entire party (which at level 3 can also raise anyone who needs it). A tank LB will boost the defense of everyone in the party.

The healer LB in particular can save the party from what would otherwise be a wipe, so it's common to hold off on other LBs until it's obvious whether that one's going to be needed or not. Then, if you get into the later part of the final boss fight and haven't needed the healer one yet, the melee DPS (or caster DPS if there are multiple adds in the final boss fight) frequently gets used to finish off the boss.

EDIT:
Ah, guess Nalia beat me to it. Well anyway, if you've been doing lower level dungeons, the healer LB is only rarely needed on those, and the melee DPS LB tends to be the most popular.

As to your other question about chains and chain bonuses: After you kill a mob, there will be a short timer, and if you kill another before it runs out, those kills are chained together. If you kill several in quick succession without breaking that chain, it boosts the amount of XP you earn for each successive kill. (There are a lot of FATEs that involve big crowds of rapidly spawning mobs. Those type are good for building up a high chain bonus.)