The President of the Court was a title given to a person who presided over a court of law. A President of the Court could oversee such legal matters as a Starfleetcourt martial.
At James T. Kirk's court martial in 2267, PortmasterStone served as President of the Court. At the start of the court proceedings, Stone asked Kirk if he had any problem with Stone acting in that capacity. Kirk replied negatively, and the court martial proceeded. (TOS: "Court Martial")
In the final draft script of "Court Martial", Stone instead referred to himself as "ChiefJudge". However, this was changed to "President of the Court" by the time the revised final draft of the script was submitted.