The Red Mage is one the starting jobs granted by the Wind Crystal.
The Red Mage in Final Fantasy has traditionally been a Jack of All trades, Master of None: It doesn't excel at magic or physical attacks, but can use both with impunity. The nature of this means that across the series the Red Mage varies from being a useful job throughout the entire game to being somewhat niche.
In III, the Red Mage falls on the second side of the scale.
Unlike in most Final Fantasy games, Red Mages in III are limited to two specific swords (denoted by their sprites in battle): The Wightslayer and the Tyrfing. Outside of those two swords, Red Mages have the same equipment as the other Mages. And because this is Final Fantasy III, the early game emphasizes magic, meaning that one or two of your characters will likely be one of those other mages anyway. While it can still be supportive up to the Fire Crystal, it's hard to recommend them afterward.
The remake is another story
In addition to gaining an extra tier of magic to use, the remake gives the Red Mage a larger range of armor and weapons, including swords, to use. Not only does this allow the Red Mage to function as a decent damage dealer until the mid-game, but it allows them to wield the swords that stack stat boosts on them.
And if you can get the Ultima Weapon + the Red Mage's job mastery item, the Crimson Vest...
Red Mages can use magic from both the White and Black magic schools. In the original, this capped at level 4 spells, but the remake bumps it up to level 5; Cruicually, this includes Raise, Curaga, Protect and Thundaga, giving the Red Mage more utility. The Pixel Remaster retains this added magic level.
The guide notes that Red Mages alone aren't sufficient enough muscle and should therefore be paired with another physical attacker, preferably one that does not use swords. Another aspect is to take care in assigning and using magic: Red Mages can use magic from both classes, but only has three slots per level. Not only that, but their black magic can't keep up during the later portions of the game. Lastly, one of the tips it mentions is the Golem Staff trick: As Gradual Petrification is more effective than one would think on common enemies, two mages attacking physically with 3-4 magic staves is enough to oneshot most enemies in a single turn.
As alluded to before, the Red Mage is a good candidate for wielding the Ultima Weapon should the player finish the Mognet questline and get the Red Mage's job mastery item. The Red Mage can wield Excalibur from Eureka, which adds +5 to all stats and UW, which adds +15 for a total of +20. However, Red Mages also get the Crimson Vest, which adds a further +10. All together, this gives the Red Mage +30 to all stats, which makes it very effective, to say the least.