The concept of the rapture is usually based on 1 Thessalonians 4:15-17 where it is argued that there is a gap of some considerable time between the events of verse 16 and those of verse 17. See below for the text:

According to the Lord’s own word, we tell you that we who are still alive, who are left till the coming of the Lord, will certainly not precede those who have fallen asleep. For the Lord himself will come down from heaven, with a loud command, with the voice of the archangel and with the trumpet call of God, and the dead in Christ will rise first. After that, we who are still alive and are left will be caught up together with them in the clouds to meet the Lord in the air. And so we will be with the Lord for ever. 1 Thessalonians 4:15-17 (NIV)

The Bible certainly does not mention such a gap. But the Bible doesn’t give a consistent detailed timetable for the end time. Indeed there are a variety of Biblical interpretations as to how the details will unfold.

The two main interpretations are as follows:

1. Those who believe that Jesus' return will be in two stages. The first stage will be a secret event when Jesus "appears in the clouds" to save Christians from the terrible things that are about to come on the earth. So the Christians are "raptured" out of the earth. The second stage will be Jesus 'second coming' (in this view the third time Jesus comes to earth). This is a public event when Jesus will bring history to a close and judge the world.
2. Those who believe that Jesus' return will be a single, public event. This view is a simpler view of events and ties in with passages such as Matthew 24:29-31, 1 Thessalonians 4:15-17, 2 Thessalonians 2:1-3 and Revelation 1:7. These all describe the return of Jesus as being in the clouds amidst trumpets, angelic activity, heavenly signs, a resurrection, and a gathering of saints (what the first view would call stage 1). This second view holds that there is no interim between Jesus gathering His own and the judgment of the world.

So it seems that the "rapture" view reads into one text quite a lot of theory as to how things might be whereas the "one event" view is a plain reading of what the "second coming" texts say together.