A trial board was a panel of individuals who tried someone in court.
Starfleet mutiny trials required a trial board of at least three command officers. In 2267, James T. Kirk initially denied Spock court martial for mutiny, claiming there were only two of them (namely himself and José I. Mendez) available. Spock reminded him that Christopher Pike was also available, but Kirk repeated his denial, as Pike was an invalid. Spock countered that Pike was still on the active duty list.
Later, during Spock's testimony, in which he confessed to the mutiny, the theft of the USS Enterprise, and the abduction of Pike, Mendez was not interested in seeing the events of thirteen years ago, but Kirk was. When Mendez claimed it was a deadlock and was about to overrule Kirk's vote, Kirk reminded him that they had not heard from Pike, the remaining member of the trial board. When asked, Pike sided with Kirk, breaking the tie. (TOS: "The Menagerie, Part I")