Here is what the Bible says about the Rapture:
1 Thessalonians 4:16-17So when Christ returns at the end of the world to judge us, the dead will resurrect, and then they and the righteous who had never died will go up to meet Jesus in the sky as He is coming down from heaven (Acts 1:9-11). That's all there is to it!
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 which 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.
If that's what the Rapture is about, then where did the idea of people being left behind after the Rapture come from? I believe it comes from a misapplication of this passage:
Matthew 24:37-42
But as the days of Noah were, so shall also the coming of the Son of man be. For as in the days that were before the flood they were eating and drinking, marrying and giving in marriage, until the day that Noe entered into the ark, and knew not until the flood came, and took them all away; so shall also the coming of the Son of man be. Then shall two be in the field; the one shall be taken, and the other left. Two women shall be grinding at the mill; the one shall be taken, and the other left. Watch therefore: for ye know not what hour your Lord doth come.
But using this passage as a support for the "left behind" idea doesn't make sense. Look at what is said about the days of Noah: those who were taken away were the evil people outside Noah's ark, while those who were left behind were the good people inside the ark. So when Jesus says that one will be taken and another will remain, it follows that the person taken will be evil, and the one who is left is good. But this is the opposite of what happens at the Rapture: it is the righteous who are caught up in the clouds to meet the Lord, not the unrighteous. So, in Matthew 24, when Jesus says that someone is "taken", it means that he or she is taken away by the flood of God's justice, and not that he or she is taken up in the Rapture.
At the end of this passage Jesus says, "Watch therefore: for ye know not what hour your Lord doth come." As we see from what Jesus says next, we are to watch in order to avoid something bad, and not in order to be ready for something good (such as the coming of Jesus so that we can go up to meet Him). The evil in the following are not left behind, but sent to hell.
Matthew 24:43-51As we can see, the context here does not concern the Rapture. Rather, this passage is a warning to be prepared for judgment, as we don't know when it will come. When Jesus returns it will be to judge the world, not to take some people away.
But know this, that if the goodman of the house had known in what watch the thief would come, he would have watched, and would not have suffered his house to be broken up. Therefore be ye also ready: for in such an hour as ye think not the Son of man cometh.
Who then is a faithful and wise servant, whom his lord hath made ruler over his household, to give them meat in due season? Blessed is that servant, whom his lord when he cometh shall find so doing. Verily I say unto you, That he shall make him ruler over all his goods. But and if that evil servant shall say in his heart, "My lord delayeth his coming"; and shall begin to smite his fellowservants, and to eat and drink with the drunken; the lord of that servant shall come in a day when he looketh not for him, and in an hour that he is not aware of, and shall cut him asunder, and appoint him his portion with the hypocrites: there shall be weeping and gnashing of teeth.
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