The Black Mage job is one of the jobs given by the Wind Crystal, and thus one of the jobs that players will recieve early. In Dungeons & Dragons terms, the Black Mage is the 'Warlock' or 'Shaman' in that they attack with offense and debuff-based magic spells.
Final Fantasy III notably uses the "Spell Charge" system, in which each "tier" of magic spell has its own number of uses. For example: Let's say a Black Mage has Fire and Blizzara. Fire is a tier 1 spell and has, let's say, 14 uses. Thus the Black Mage can use Fire 14 times. Let's say that the Black Mage also has Thunder, which is also a tier 1. Then both spells can be used, but both count towards that same 14 total. Now, let's say Blizzara is a tier 3 spell. It has 4 uses. These four uses are entirely seperate from the 14 uses that Fire and Thunder have. The usage of one does not impact the other in any meaningful way.
Unlike the Final Fantasy remaster on GBA, the III remake retains the same spell charge system that the original used.
The remake didn't directly impact the Black Mage in particularly meaningful ways, for the obvious reason of there not being much that can fundamentaly change about it. Instead, the Black Mage got inadvertedly nerfed as a result of the instand-kill/status spell success rates being flattened: Whereas after a certain point, Black Mages could reliably take out opponents with those spells, they're far less likely to hit now. This is tempered by the 3D remake only having 3 foes in a fight per battle, but it's still worth noting. As a result, Black Mages in the remake are better off focusing on general damage spells.
Black Mages can use Black Magic up to level 7. While it's not the only class to use Black Magic, it is the only non-endgame job to use magic at that high of a level, ensuring that it remains in use until late in the game.
In a recurring theme, defensive characters such as Knights are recommended due to the Black Mage's weak defense. The Future Press guide notably mentions that it's possible to make a offense-oriented party out of two black mages and a Dark Knight, although without a White Mage, it'll be painful to heal up. As far as spells go, hitting enemy weakness is the priority and whatever lacks a weakness can be attacked with Bio: A non-elemental magic attack that deals good damage. Magic-casting items such as staves and rods are also useful for conserving MP, even if they only go up to level 4.