You could say that the origin of trance comes from hostility. The hostile energy of enemy attack induces trance, that’s why being attacked by your allies can’t cause trance. By the way, the trance gauge appears both in battle, and in the status menu, which is under the main menu.
Zidane’s trance replaces Skill into Dyne. Dyne is a whole set of moves which are learned when Zidane learns his regular skill moves. Dyne also uses little MP. He can cause a lot of damage to one enemy or he can cause a massive explosion causing damage to all enemies.
Vivi’s trance changes from black magic to double black magic. Instead of using just one magic, Vivi uses any two twice. This happens in one turn but uses MP twice as well.
Garnet’s trance changes summon to Eidolon. It’s pretty much the same thing. Garnet still summons a summon.
Steiner’s trance does double damage in his normal attacks. He still has sword arts and he also has sword magic. Steiner can attack the enemy and then it does a specific magic against the enemy as well. He only can do this when Vivi is on the team.
Freya’s Trance turns Jump into well, jump. Instead of jumping up and coming down once, she stays up there until her trance is over and unleashes spears onto her opponents every turn.
Eiko’s trance turns white magic into double white magic, just like Vivi’s. She can use two magic. But her magic can cure, and cause Holy damage. She has different magic than Garnet but has only 4 summons.
Amarant’s trance doesn’t change. But, his flair skills can hit multiple times than his normal Flair.
Quina’s trance turns eat into Cook. It’s pretty much the same thing, but you have more of a chance to eat it. You also don’t have to weaken it more.