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")
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...")