(covers information from several alternate timelines)
A field commission (also called a (battle)field promotion or battlefield commission) was a permanent promotion given to enlisted or commissioned soldiers for outstanding leadership on the field of battle.
In 2155 in the mirror universe, Jonathan Archer reminded Admiral Black that he had the authority to give him a battlefield promotion. When Black reminded him that they had no starship to give him command of at the moment, Archer requested the USS Defiant. (ENT: "In a Mirror, Darkly, Part II")
In the alternate reality, Captain Christopher Pike gave a field promotion to Commander Spock, to acting captain, and Cadet James T. Kirk, to first officer. After Spock was revealed to be emotionally compromised by the destruction of Vulcan, he resigned his command, per Starfleet Regulation 619, which automatically field-promoted Kirk to acting captain. Kirk was formally recognized by Starfleet as the captain of the USS Enterprise, while Spock resumed his original rank and position as first officer. (Star Trek)
Martok, who served as a laborer in the Klingon Defense Force, received such a battlefield commission from General ShiVang due to his actions in defending the general's flagship from the Romulans. (DS9: "Once More Unto the Breach")
In Starfleet, starship captains were given much leeway in interpreting Starfleet regulations and assigning crew positions. Wesley Crusher was field-promoted from "acting ensign" to full ensign by Captain Jean-Luc Picard in recognition of Wesley's academic accomplishments and sacrifice in a rescue mission saving William T. Riker, Deanna Troi, and Lwaxana Troi. (TNG: "Ménage à Troi")
In 2366, Vice Admiral J.P. Hanson gave Commander Riker a promotion to the field commission of captain. Riker chose to resume his previous rank and position as first officer when Captain Jean-Luc Picard was recovered. (TNG: "The Best of Both Worlds", "The Best of Both Worlds, Part II", "Family")
These types of promotions were done without the approval of the organization's central authority, but by commanding officers who operated outside communication with authority. For example, Captain Kathryn Janeway, after having "invited" Chakotay and the other Maquis to become part of the crew of USS Voyager, with Chakotay serving as first officer in 2371, she granted Thomas Eugene Paris a field commission of lieutenant and an assignment at the conn. (VOY: "Caretaker")