A supervisor (or supervising officer) was someone who had been given low-level management responsibility.
In 2154, Piral, Trenia, and Jaina were Xindi supervisors. (ENT: "Damage")
In 2370, mirror Odo was supervisor of the ore processing center on Terok Nor. (DS9: "Crossover")
Vorta who commanded a unit of Jem'Hadar in the field were known as "field supervisors". In 2372, after meeting Captain Benjamin Sisko for the first time, the Vorta Weyoun noted that the Starfleet officer was required reading for all Vorta field supervisors. (DS9: "To the Death")
In 2375, Yost held the title of supervisor. He commanded a ship which was part of Renovation Team Nova. (VOY: "Gravity")
In the same year, Lieutenant Douglas was a supervising officer at Starfleet Medical. She reported to Commander Hilliard and one of her subordinates was Ensign Weldon. (DS9: "When It Rains...")