Force ghost
In the Star Wars universe, a Force ghost (also known as Force Spirit) is the apparition of a deceased Jedi who has become one with the Force.
During the original trilogy, it was believed that all Jedi who died became one with the Force, their bodies vanishing, and then could return as apparitions at will, so that they could help those in need of their guidance. This would happen even to fallen Jedi who repented at the end of their lives, such as Darth Vader and the Jedi Ulic Qel-Droma (from the Expanded Universe). In the former case, it is unclear if Vader's body actually vanishes, since Luke Skywalker is seen putting together a funeral for his suit, but it is assumed that the organic part of his body has dissapeared.
Many questions arose during The Phantom Menace, when Qui-Gon Jinn's body did not vanish after his duel with Darth Maul, and instead was burned on a Jedi funeral pyre. In Revenge of the Sith it is made known that "becoming one with the Force" is not in fact what happens to every Jedi, but only a recently discovered ability. Qui-Gon's spirit guided Yoda and Obi-Wan, and revealed to them the secret of retaining one's individuality after becoming one with the Force. Presumably, Darth Vader did not learn of this ability, which explains his surprise when Obi-Wan's body disappears when he is slain on the Death Star in Episode IV.