Well I guess I have to admit my geekness. From the Memory Alpha Star Trek web site. The first paragraph has the meat of what it is/was.
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 Klingon Neutral Zone. The starship would receive a distress signal from the SS 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 is 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".
During the filming of The Wrath of Khan, some people voiced concern at the notion of Kirk having "cheated" to pass the test. However, Nicholas Meyer defended the notion, saying it revealed an aspect of Kirk's character, and that the film, or Kirk, shouldn't be restricted by "television mentality".
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." (Star Trek II: The Wrath of Khan)
The term "Kobayashi Maru" may be a slang term for any hopeless situation in the 23rd century, at least in Starfleet Culture. Leonard McCoy considered his and James T. Kirk's imprisonment on Rura Penthe to be a "Kobayashi Maru", and told Kirk as much their first night at the penal mine. (Star Trek VI: The Undiscovered Country)
