Black Belt is one of the late-game jobs in Final Fantasy III. It can be considered an upgrade to the Monk job, mirroring the Master class upgrade in the original Final Fantasy.
In the original Final Fantasy III, Black Belt came at the Water Crystal. Much like Monk, it is a physical powerhouse once it gets some job levels and the like under its belt. This makes it useful during the opening sections of the Surface section, especially against Goldor who nulifies magic. Later dungeons that require more specialized jobs do hurt its effectiveness though, but it otherwise remains a solid choice until the Eureka jobs.
In the remake, Black Belt was moved to the Earth Crystal. In an interview with the game's developers, it was revealed that this was to balance out the types of jobs: With Sage and Ninja also being moved to the Earth Crystal, that meant that the crystal only gave one, single physical job. The developers wanted to not make the Phys/Magic ratio so lopsidded, but of the Water Crystal jobs, the Black Belt was the only one who could be moved without much issue.
Still, this means that attempting to use Black Belt is harder due to the need to build up job levels to make physical attacks more effective. By the time the player gets Black Belt, they'll likely be on the way to the Crystal Tower, meaning they'll have to backtrack (unless they were savy enough to get the Earth Crystal prior to Doga's Grotto, which is possible, but doesn't change the need for grinding). Black Belt can still be powerful though and it's got one of the highest vitality stats in the game, making it useful for level grinding.
In the Pixel Remaster, in addition to retaining Charge, the Black Belt was given the Kick command. This is one of the few commands added in the PR that did not initially appear in the 3D remake
In the 3D remake at least, Black Belts have access to the Martial Arts passive: When the Black Belt is not equipped with any weapons, they use a special damage calculation formula: ((base) Strength + Job Level) x 2. Hand levels further improve their strength. This means that while Black Belts will need hand weapons early on, the higher job levels eventually allow them to truly make the most of their monsterous attack power.
Charge allows you to use a turn to increase your Attack. The Command can be used up to three times: The first two times will increase your power thusly while the third causes a "backfire" and damags your character. Unlike Geomancer, the backfire mechanic has remained even in the later versions.
The Kick ability is a signature ability for Monks starting with Yang from Final Fantasy IV. It is a physical attack that hits all enemies on the screen. This ability is exclusive to the PR version.
As one would expect, this makes it very useful when stacked with Charge. The Black Belt can prove to carry a player even into the endgame just through this change alone.
As with Monk, Black Belts won't have many options for elemental attacks, so a Black Mage comes in handy. Additionally, by this point the party has jobs that can make the Black Belt hit even harder (Viking to lower enemies' defenses, Bard to buff the party's phys attack). However, the need for high job/hand levels + limited selection of claws means that it may be sub-optimal to switch from Monk to Black Belt, much less consider fielding more than one at a time.