(covers information from several alternate timelines)
A provisional (or brevet) officer was a member of a military or service organization who had received a temporary commission, field commission (or field promotion) to hold a rank as a commissioned or non-commissioned officer in that organization.
Starship captains were given much leeway in interpreting Starfleet regulations and assigning crew positions. Wesley Crusher was given a field appointment as an "acting ensign" by Captain Jean-Luc Picard. (TNG: "Where No One Has Gone Before") The captain later field-promoted him to full ensign, recognizing Wesley's academic accomplishments and sacrifice in a rescue mission saving William T. Riker, Deanna Troi, and Lwaxana Troi. (TNG: "Ménage à Troi", "The First Duty")
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)
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 awarded provisional ranks to several capable officers of the Maquis vessel Val Jean in 2371 when USS Voyager became stranded in the Delta Quadrant. (VOY: "Caretaker")
In 2366, Vice Admiral J.P. Hanson gave Commander William T. Riker a field promotion to the grade 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")
On the USS Valiant in 2373, the seven regular officers were killed in a skirmish with a Cardassian battle cruiser leaving the Red Squad cadets to run the ship. Many were field promoted by Tim Watters to various ranks after he assumed command of the vessel. (DS9: "Valiant")