Dismissal or firing was an edict given by supervisor ending an individual's employment from their occupation.
When the USS Enterprise was being pulled from the orbit of Gamma Trianguli VI by Vaal in 2267, Captain James T. Kirk, who frustrated with the news he was receiving from Commander Montgomery Scott regarding his inability to resolved the situation, warned him that "if you can't get those warp engines working you're fired." Later, when Scott reported that his attempts to escape orbit bought the ship and crew another hour, at the expense of blowing out almost every system on the ship, Scott admitted that there was nothing left to try, concluding that "I guess you'll have to fire me, sir." Kirk, thereafter, grimly told Scott "you're fired." Later still, Scott was able to use the ship's phasers to destroy Vaal, which allowed them to escape Vaal's tractor beam, Kirk quickly told Scott, "you're re-hired." (TOS: "The Apple")
In 2370, when Quark's new waiter Pel's business sense began making Rom jealous, Rom kept attempting to fire the new employee, first after Pel admitted that they didn't plan on being a waiter forever, then after Rom felt Pel had insulted Quark, as they proposed that Quark's lobes for business were only part of the reason why Grand Nagus Zek chose him to conduct business with the Gamma Quadrant. (DS9: "Rules of Acquisition")
In 2373, Zek fired Quark from his job as First Clerk after it was revealed he had maneuvered Zek into thinking his mother and Zek's lover Ishka was manipulating him, instead of warning him about her, which was the reason Quark was given the job in the first place. (DS9: "Ferengi Love Songs")
In 2380, Fletcher was fired from his posting on the USS Titan after only six days when he was caught throwing garbage into the ship's warp core. (LD: "Terminal Provocations")