Kobayashi Maru
The Kobayashi Maru was a Class-III neutronic fuel carrier and a component of the Kobayashi Maru simulation, a no-win-scenario at the Starfleet Academy.
The home port of the vessel was Amber, Tau Ceti IV. The ship's master was Kojiro Vance.
According to the scenario, the Kobayashi Maru was located in Gamma Hydra, Section Ten, nineteen periods out of Altair VI. The vessel had struck a gravitic mine, and its hull was breached. The ship had lost all power, life support was failing and its crew had sustained many casualties.
The Kobayashi Maru scenario was an infamous no-win scenario that was part of the curriculum for command-track cadets at Starfleet Academy in the 23rd century. It was primarily used to assess a cadet's discipline, character, and command capabilities when facing an impossible situation.
In the scenario, a cadet was placed in command of a starship on patrol near the Organian Neutral Zone. The starship would receive a distress signal from the Kobayashi Maru, a civilian freighter that had been disabled in the zone after having struck a gravitic mine. If the cadet chose to enter the neutral zone in violation of treaties, the starship would be confronted by three Klingon K't'inga-class battlecruisers. The test was considered a no-win scenario because it was impossible for the cadet to simultaneously save the Kobayashi Maru, avoid a fight with the Klingons, and escape from the neutral zone with the starship intact.
There were likely several variations of the Kobayashi Maru scenario that existed at various times. It's probable that the Academy instructors periodically retool the scenario to fit current events in the galaxy.
In the 2250s, James T. Kirk became the first (and only known) cadet to ever beat the no-win scenario. After taking the test and failing twice, Kirk took the test a third time after surreptitiously reprogramming the computer to make it possible to win the scenario.
Kirk got a commendation for "original thinking", and later commented wistfully that his stunt "had the virtue of never having been tried." Kirk would later defend his "cheating" by arguing that he didn't believe in the no-win scenario. Ironically, Kirk also defended the test itself by suggesting "how we face death is at least as important as how we face life".
In 2285, Kirk, then an admiral serving as an instructor at the Academy, supervised Lieutenant Saavik's performance in the Kobayashi Maru scenario. Former USS Enterprise crew members Spock, Sulu, Uhura, and McCoy participated as "actors" in the simulation. Saavik's performance was predictably dismal; as Kirk observed, "She destroyed the simulator room and [the crew] with it."
The term was later generally applied by Dr. McCoy to define an unwinnable or potentially fatal situation. He considered his and James T. Kirk's imprisonment on Rura Penthe to be a "Kobayashi Maru".
A similar simulation was later used in the 24th century. It involved a damaged Ferengi ship and Romulan warbirds instead of Klingon battlecruisers, and was performed on the holodeck.
http://memory-alpha.org/en/images/4/41/Kobayashi_Maru_data.jpg