The Picard Maneuver was a type of tactical maneuver that can be made by warp capable vessels. The maneuver was first executed by the USS Stargazer, under the command of Captain Jean-Luc Picard in 2355 during the Battle of Maxia. Afterwards, Starfleet began referring to the maneuver as the Picard Maneuver. The Picard Maneuver was often one of desperation, used to fool an attacking ship just long enough to gain a critical advantage. (TNG episode: "The Battle")
An officer who decided to use the Picard Maneuver would first order a sensor bearing to get the exact position of the opposing vessel. Then, the officer would order his or her helm officer to drop into high warp for a few seconds to close the distance between their ship and the opposing vessel. Because of the physics involved, the light from the previous position will not have moved, and to the opposing vessel the ship will briefly appear to be in two places at once.
The defense from this maneuver involved using a ship's sensors to detect a sudden compression of trace gases which indicated another vessel was about to use the Picard maneuver, and to use a ship's tractor beams to capture the other vessel and limit it's field of fire. (TNG episode: "The Battle")
The Picard Maneuver is known to have been used at least twice, and in both instances it was executed by Captain Picard.
In 2355 the Stargazer was in the Maxia Zeta star system when it encountered what was at the time an unknown vessel. This vessel proceeded to attack the Stargazer. During the course of the battle, the Stargazer was severely damaged. Captain Picard ordered his helm to obtain a bearing on the attacking vessel, and upon receiving that bearing ordered his ship to jump into high warp for several seconds. He then ordered the ship to come to a full stop near the unknown ship. As a result, the Stargazer briefly appeared to be in two places at once - and the unknown ship fired where the Stargazer had been. The Stargazer fired phasers and six photon torpedoes at the ship, which exploded. Picard had to abandon the Stargazer, and he was court martialed but ultimately acquitted for the loss. Picard was honored with the Grankite Order of Tactics for coming up with what would become known as the Picard Maneuver at the battle.
Nine years later, in 2364 the Ferengi DaiMon Bok used thought makers to manipulate Picard into believing that he was reliving the battle. Picard, believing the USS Enterprise-D to be the attacking Ferengi starship once again used the Picard maneuver in an attempt to attack the Enterprise. However Lieutenant Commander Data was able to construct a defense which prevented Picard from attacking his own ship. (TNG episode: "The Battle")
Amongst the crew of the USS Enterprise-D the Picard Maneuver was also the light hearted nickname given to Captain Picard's characteristic tug of his uniform top every time he got up. (TNG novel: Vendetta)
In this version of the mirror universe, Picard's counterpart in this universe, Lieutenant Jean-Luc Picard, utilized a similar maneuver to escape Cardassian attack, shortly after taking command of the ISS Starbreaker. In this case too, the maneuver was improvised on the spot. (TNG comic: "Mirror Images, Issue 3")
In this alternate version of the mirror universe, Captain Jean-Luc Picard, utilized a similar maneuver to escape Ferengi attack. In this case too, the maneuver was improvised on the spot. (TNG novel: Dark Mirror)
In another timeline, Picard improvised the maneuver, used in a conjunction with a saucer separation, to counter the numerical superiority of an Alliance fleet, the unprecedented tactic forcing the aliens to retreat. (TNG comic: "Mirror Broken, Issue 5")