The Tactical Officer is an officer stationed at the console that controls a ship's weaponry and shields.
On Starfleet vessels, the tactical officers job is to maintain weapon and shield readiness along with executing firing solutions. These solutions can range from a low intensity phaser beam or to plot a torpedo firing pattern. Security chiefs frequently doubled as tactical officers, though the two roles weren't required to be filled by the same person. (TNG: "Pen Pals", "Masks")
Aboard Federation starships in the mid-23rd century, these duties were handled, at least to a large degree, by the helm officer and sometimes by the navigator. (TOS: "The Corbomite Maneuver")
On a Klingon vessel, this crewmember is called the Weapons Officer.