A Director's cut is a specially edited version of a movie that is supposed to represent the director's own approved edit of the movie. It is often released some time after the original release of the film, where the original release was released in a version different from the director's approved edit. 'Cut' is synonymous with 'edit' in this context.
Movies which have a director's cut version include:
Another concept, recently popularized by director Peter Jackson, is that of an extended or special edition, which he used for special releases of his Lord of the Rings movies. While Jackson considers the theatrical releases of those films to be his final 'director's cut', the extended cut was made so that fans of the material could see nearly all of the scenes shot for the script, but that were cut for theatrical running time or other reasons. Other examples of extended/special editions include James Cameron's lengthier cuts of Aliens and Terminator 2, and Luc Besson's Version Intégrale cut of Léon.
Less frequently, video games will receive rereleases with added material under a director's cut label. One of the earliest games to use this concept was Resident Evil for the SonyPlayStation.