The Monk job is one of the jobs granted by the crystal of Wind, and therefore is one of the starting jobs of Final Fantasy III. Monk is an alternative option as far as physical attackers go, albeit with more extreme pros and cons.
Monks are a low-investment class. Early on, they need nunchucks and claws in order to augment their damage. However, as their (job) level increases and their hand stat grows (more on that elsewhere), the more powerful their attacks get. Eventually, their bare hands become one of the strongest weapons in the game (though claws are still helpful to augment them further).
The catch is that Monks have rarer equipment. Mostly gis and such. As a result, the class can be glass cannon-ish since it needs to go further stretches to reach better equipment. Still, if you can handle it, Monks can definitely dish as much as they take and prove themselves a worthwile job to use, at least early on.
The remake largely keeps Monk the same. However, the nunchucks and tonfas that it previously had are now gone, leaving only the claw weapons for use. This can make the early stages harder since your hand stat is likely to be low.
The Monk's unique command in the 3D remake is Retaliate. Going before everyone else, the Monk does a gesture. Over the course of the next turn, if the Monk is struck with an attack, they will counterattack.