(covers information from several alternate timelines)
A referee was an individual who had authority during a sporting competition. A baseball referee was called umpire.
A referee was part of a water polo match. Such a referee was present in the recording of such a match watched by Jonathan Archer in 2152. (ENT: "The Catwalk")
In 2153, Rianna Mayweather claimed she had played referee between her two brothers for a long time. (ENT: "Horizon")
As a young woman, Norah Satie would debate different topics with her older brothers over a specific topic every night at dinner, with Aaron Satie being the referee. He would make use of a stopwatch to teach his children brevity. (TNG: "The Drumhead")
In 2342, one of the referees made a controversial decision against the parrises squares team of M'Kota R'Cho, calling a penalty. R'Cho subsequently strangled the referee. (VOY: "Year of Hell")
At the 77th Emperor's Cup of sumo, according to Chakotay, Takashi's knee broke the sand, so the referee gave the match to Kar-pek. (VOY: "Latent Image")
In 2374, Chakotay told B'Elanna Torres he was tired of playing the referee for every confrontation between her and Seven of Nine. (VOY: "Message in a Bottle")