The Exorcist is a mind-bending job that supports allies by undoing things that have happened. It has great MP and Mind and generally well-rounded stats elsewhere, save for its poor Strength. As such, it can function as a competent healer, but not a good fighter despite its affinity for swords.
The Exorcist's main gimmick is that it can revert a character's HP, MP and BP to a previous turn's value, effectively healing them. Unlike healing spells though, the Exorcist's abilities only work on single targets and not the whole party, but there's no denying their incredible potential.
Specialty: Undo Trois
When using Undo HP, Undo MP or Undo BP, you can choose whether to revert to the HP/MP/BP value from two turns ago or three turns ago (in addition to one turn ago).
During hectic battles, a lot can happen in a few turns, so it's great to have more leeway. Extending the range of past turns also opens up many more strategical options.
Level 1: Undo HP
You're probably all too familiar with this ability, having witnessed it during your battle with Geist. In your hands, you can use it to heal or revive allies or punish enemies that try to heal.
Level 2: Undo Action
The game's description is slightly ambiguous; what this does is completely negate all effects of one enemy move. In a sense, it's a reverse Utsusemi that can also evade magical attacks, ailments and debuffss--which makes it invaluable. It works great with Prescience as well.
Level 3: Steady MP Recover
For battles that take longer than 1 turn, this ability is ace. Besides Exorcist, other jobs that rely on MP, such as healers or spell-casters, will love it as well. Although if you're using magic, you're better off using MP Free in a Pinch as a ghost.
Level 4: Purgation
This is basically Dispel from Bravely Default, but it's no longer magic and thus cannot be cast on a party with Group-Cast All. Still, it's good for removing negative effects from your party members and it's one of the few ways to cure Stop. Also good for removing buffs from enemies.
Level 5: Undo MP
MP is more expensive compared to HP, so this ability has its uses. However, since it requires expending what equates to two turns, it's usually too unwieldly to use. Instead, put on Steady MP Recover for your Exorcist and MP Free in a Pinch for your ghost mages.
Level 6: Undo BP
This ability is expensive as heck and generally too much of a hassle to use, but it has a lot of strategic merit, especially when Undo Trois is available.
For instance, you could have an attacker save up 3 BP, then Brave the maximum amount of times over two turns, dropping them to -3 BP. Afterwards, your Exorcist can instantly return the attacker back to 3 BP for a fresh assault.
Level 7: Undo Trois
If you're planning to use the Exorcist's abilities in a different job, you should definitely equip this.
Level 8: Eradication
Not a reliable move by any means, but possibly your Exorcist's best way of defeating foes. Also useful for taking down foes who are far, far stronger than you. Can be combined with Lucky Day for an improved success rate.
Level 9: Ectoplasm
This allows your Exorcist to continue fighting on even after sustaining a KO. Just keep in mind that Exorcism doesn't count as magic, so your Exorcist will have to change roles when they're a ghost. Since ghost is the same as a KO, don't use this if you have three other ghosts.
Level 10: Glossolalia
This ability provides another way for your Exorcist to continue assisting while they're a ghost. Additionally, if you can somehow co-ordinate with your friends, you could potentially have all of the best spells all under one roof!
By the way, the spells available with be unmodified by your friends' Spellcraft, but you can use Spellcraft yourself if you have the ability.
Level 11: Auto Undo
The cost is enormous, but as they say, you get what you pay for--which in this case is the ability to become nigh invincible. Of course, you can still get KO'd by massive damage, so it's a good idea to give this to a sturdy character, such as a tank.