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...
The concept of a "do everything" class goes all the way back to Dungeons & Dragons with the Cleric job, with the term "gish" being used to describe sword & sorcery characters. In particular, Red Mages tend to be considered the "FF" equivalant of Bards in D&D, at least originally.
The Red Mage has been around since the first Final Fantasy and made off-and-on again appearances throughout the series. Unlike most other jobs, it got few "signature" characters, although Square-Enix has since seemingly settled on Alisae from Final Fantasy XIV as their go-to Red Mage representative.
In the Perfect Guide Volume 1, it is suggested that Red Mages specialize in magic that existed prior to the polarization of Black and White Magic. Red Mages came about out of a sort of desire to follow the original magic. Red Mages are also heavily associated with Sasune: In the guide, it is said that it was once customary for Red Mages to ascend the throne while a soldier in the remake states that all Sasune soldiers strive to become Red Mages.
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.