A search and rescue operation, also known as a rescue mission, rescue operation, or search operation, was a mission with the goal to save someone from a situation or another party. They were carried out by rescue teams.
While under the influence of trellium, T'Pol threatened to kill Jonathan Archer, believing the mission was to eliminate the Vulcan crewmembers aboard the Seleya. Archer tried to explain her that they were on a rescue mission and pointed to the stun setting of their phase-pistols. (ENT: "Impulse")
In 2372, Tom Paris was charged with leading a rescue mission to bring back Kathryn Janeway, B'Elanna Torres, and Tuvok from a Mokra Order prison. (VOY: "Resistance")
In 2375, the crew of Voyager started a rescue mission for Tuvok, Tom Paris, The Doctor, and Noss, who were trapped in a subspace sinkhole. (VOY: "Gravity")
In 2384, Vice Admiral Kathryn Janeway ordered Commander Tysess to launch a search and rescue operation for the USS Protostar crew. Despite Tysess objecting that it could take months to search the distance that the ship had crossed, Janeway wanted the young crew found if they were still alive because they had just saved Starfleet. Ultimately, the crew managed to make their own way to Earth a month later. (PRO: "Supernova, Part 2")