The White Mage is one of the five or six jobs that is available after encountering the Wind Crystal. As the opposite of the Black Mage, the White Mage focuses primarily on healing and buffing the party or getting rid of status effects. That's not to say that the job has no offensive skills though, as it is its the wielder of the Aero family of spells. As in Final Fantasy II, curative magic has the ability to deal damage to the undead.
As a healer, the White Mage is generally useful throughout the game. They are heavily associated with the staff weapon type - while Black Mages can use early-game staves, they cannot equip late-game staves, which boost the mind stat.
The remake did not affect White Mages in a direct way, although their usefulness in the early game is increased in an indirect fashion: Staff-related magic has been buffed and the limit of three enemies per battle means that multi-target magic is not as useful. Therefore, they are a boon for getting through the Tozus tunnel and Nepto Shrine, dungeons that require the party to become mini resulting in magic attacks not being effective.
White Mages can use up to Level 7 White Magic, a feat that no job other than the endgame magic classes can accomplish. This ensures that a White Mage can remain useful for most of the game.