I don't understand what a Limit Break is for, especially the Cure version. Please coach me in the use of this function.

I have only run the first two dungeons so far, and I've had no trouble keeping my party alive. I have the icon on my action bar, and I understand that we get a "one-bar" LB when fighting regular bosses, and we get a "two-bar" LB when we reach the final boss, thus boosting the effect of the Limit Break. What I don't understand is:
  1. What does a Limit Break do?
  2. When do I want to use one?

I expect that the post-30 dungeons will be hard on Healers, and that I will need to use the Cure Limit Break at some time during one of those dungeon-runs. So..., please enlighten me.