A duty station was the station that a duty officer, that is, one who was on-duty worked from.
During the 2270s, when the USS Enterprise became caught in V'ger's tractor beam, and Admiral James T. Kirk and Captain Will Decker contemplated what to do next, Kirk first had Decker order the duty personnel to "resume duty stations." (Star Trek: The Motion Picture)
Later, during the same mission, the ship's computer announced the current crew status, noting that 172 were at duty stations, 248 were off duty, and 11 were in sickbay for minor conditions. (Star Trek: The Motion Picture)
"Mr. Adventure" considered his assignment at Old City Station to be "the worst duty station in town" in 2285. (Star Trek III: The Search for Spock)
While under the influence of the Bok's thought maker in 2364, Captain Jean-Luc Picard had returned to his quarters aboard the USS Enterprise-D to rest after ordering Doctor Beverly Crusher and Counselor Deanna Troi to their duty stations, as they were later reminded by Commander William T. Riker. (TNG: "The Battle")
In 2365, after becoming trapped in Nagilum's "hole in space", the Enterprise-D encountered what was believed to be a Romulan D'deridex-class warbird, Commander Riker ordered the ship to go to red alert, prompting Dr. Katherine Pulaski to state that she was going to return to her duty station. (TNG: "Where Silence Has Lease")
After the Enterprise-D received a distress call from the Ohniaka III Outpost in late-2369, Commander Riker ordered the ship to red alert, and requested that "all personnel report to duty stations." (TNG: "Descent")