Fleet commander was a position in the Vulcan and Klingon militaries, in charge of overseeing fleets on the front lines. Vulcan fleet commanders reported directly to the Vulcan High Command.
In 2154, Minister Kuvak contacted the fleet commander in order to withdraw the Vulcan ships in the Battle of Andoria. (ENT: "Kir'Shara")
By 2367, Lursa and B'Etor had the loyalty of at least three Klingon fleet commanders, according to Gowron, who attributed it to fear. (TNG: "Redemption")
In an alternate timeline 2401, Tuvok was a fleet commander in the Vulcan Defense Forces and fought in the Vulcan War. He was listed as the Vulcan leader in the summary for this war in the intelligent briefing prepared for President Annika Hansen. (PIC: "Penance")
Fleet commander is presumably equivalent to a fleet captain or a fleet admiral.
A deleted scene from Star Trek Nemesis revealed that Romulans also had fleet commanders. The scene was set shortly after Shinzon's coup, and discussed how the fleet commanders were getting nervous at being in the dark.