The Vulcan greeting Live long and prosper is the most recognizable catchphrase of the Star Trek franchise, comparable to May the Force be with you of Star Wars fame. In The Simpsons episode Marge vs. the Monorail, Mayor Quimby, in yet another of his misguided attempts to reference pop culture, mistakenly uses the wrong catchphrase when greeting episode guest star Leonard Nimoy.
This instantly identifiable phrase is used by the Vulcan people as both a greeting and a farewell (sometimes replied to with Peace and long life), and is almost always accompanied by the equally famous Vulcan salute. The Vulcan salute is a palm-forward gesture with the fingers and thumb held in a W shape, with the first two and last two fingers each held together. It can be difficult for some people to perform correctly because the ring finger lacks a separate muscle apart from the pinky finger (Wikipedia). Notably, William Shatner is reportedly unable to perform the salute, and the character Zefram Cochrane had difficulty returning the gesture to the Vulcans in Star Trek: First Contact.
Both the phrase and the salute made their first appearance in the original series episode Amok Time, the fifth episode of the second season, in which Spock must return to Vulcan for Pon Farr, the Vulcan mating ceremony. The phrase was scripted (and first spoken not by Spock, but by T'Pau), but the salute was improvised by Leonard Nimoy, who thought the Vulcans should have a unique gesture to go along with it. He based it off of a traditional Jewish gesture of blessing, which looks very similar but is performed with both hands. This is only one of several aspects of Vulcan culture that can be credited to Nimoy's inventiveness, along with the non-lethal Vulcan nerve pinch.
Other notable appearances of the popular phrase include First Contact and The Wrath of Khan. Live long and prosper were the first words spoken by Vulcans to humans at their official first contact on April 15th, 2063 (although the Vulcans had been monitoring Earth long before then). Fittingly, they were also Spock's last words to Captain Kirk before dying of radiation sickness, sacrificing himself to save the ship at the end of Wrath of Khan. Spock, of course, would later return from the dead in The Search for Spock.
It should be noted that Live long has special meaning for the Vulcan people, as Vulcan lifespans are at least twice that of Humans.
In the original Vulcan, Live long and prosper is pronounced Dup dor a'az Mubster. (Star Trek: The Motion Picture). Another well-known Vulcan saying is Kol-Ut-Shan, Infinite diversity in infinite combinations, which is represented by a triangle and a circle, forming what is fan theorized to be an image of the Vulcan sun rising over Mount Seleya. The IDIC symbol is used as a decoration on many Vulcan objects, perhaps most visibly on the Vulcan baseball team's uniforms in the Deep Space Nine episode Take Me Out to the Holosuite.