The word "rapture" comes from the Latin verb "rapiemur" which means "carried off" or "caught up". It was used in the Latin Vulgate (about 405 A.D.) translation of 1 Thessalonians 4:17, which is the primary biblical reference to the event in question.
The Rapture is a term most commonly used to describe an event in Christian eschatology (study of the end times) whereby all true Christians are taken from Earth by God into Heaven before other events associated with the end of the world take place. In Christian circles this is known as a pre-trib doctrine, because the rapture rescues the faithful from Earth before the tribulation, rather than after (post-trib), as some other Christians believe. According to the doctrine, the resurrection of the dead will occur at the same time.
The popularization of the term is associated with teaching of John Nelson Darby and the rise of premillennialism and dispensationalism in the U.S. at the end of the 19th century.
This teaching revolves around the scripture passage in Matthew 24:40-41 which explains that "two shall be in the field; the one shall be taken, and the other left" (compare Luke 17:34-36), and also 1 Thessalonians 4:17 which teaches that believers will "be caught up together...in the clouds to meet the Lord Jesus Christ in the air").
In Corinthians 15:51-52, the Apostle Paul writes: "Behold, I show you a mystery; We shall not all sleep, but we shall all be changed, in a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed."
In Thessalonians 4:13-18, Paul writes: "For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first: Then we who are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord."
In Philippians 3:20-21, Paul writes: "For our conversation is in heaven; from where also we look for the Saviour, the Lord Jesus Christ: Who shall change our vile body, that it may be fashioned like unto his glorious body, according to the working whereby he is able even to subdue all things unto himself."