The gauge is indeed shared among the party, and the LB will have a different effect depending on the class/job of the person who uses it.

A healer's LB will give a massive heal to the entire party (including Raise to all needing it if it's a LB3). If things are going badly, it can save a run that would otherwise have wiped, but is usually not needed if the run is going well.

A tank's LB gives the whole party a damage reduction buff. It could be useful for a boss that has a single massive-damage AoE hit, but such attacks usually only come if you've failed the preceding mechanic, so this is another one whose main use is rescuing the run from a near failure. (And if that preceding mechanic is a DPS check, using a DPS LB to avoid failing is generally better than using a tank LB to survive it).

Damage dealer classes/jobs, unsurprisingly, have an attack as their LB, though the type of attack varies by class:

DoM DPS jobs have an AoE attack as their LB. A few places have a swarm of adds in your final boss fight and this can help to clear them out.

DoW ranged DPS jobs now also have an AoE attack for a LB, but theirs is directional, hitting all mobs that lie in a straight line in front of them.

Melee DPS jobs have a single target attack for their LB. Of the three, it has the highest attack power. Since most dungeons end with a fight where the primary goal is killing a single boss, this melee LB is probably the most often desired and used LB of all of them. If (judging from your avatar character) you're sticking with Dragoon, then using the team's Limit Break will often be your responsibility, particularly if the other damage dealer in the party is a ranged or caster class.

As for when to use a melee LB, in general wait until the final boss of the dungeon and then once the LB gauge is completely filled, fire away. If the fight has a difficult DPS check phase, however, then you may want to try to get it in then. Also there may be time to do a level 1 LB early in the dungeon and still have time to fill the gauge all the way back up again in time for the final boss, but unless an interim boss is particularly difficult, I'd only do that if you're confident you'll have time for it to fill back up. (The tendency lately of people to rush through dungeons doing the minimum necessary makes this opportunity come up less often than it used to.)