The rapture is a popular, but false doctrine. It was not taught by Jesus, the disciples, or any of the reformers or early church fathers. It is a teaching that has come about relatively recently (within the last couple hundred years) and popularized by the Left Behind series of books and movie.
People get confused because they don’t know the true state of the dead, and also because they are misinterpreting the 70 Weeks prophecy in Daniel 9 (see What happens when we die? and Daniel’s 70 Weeks for more on this).
When Jesus returns to Earth, it’s going to be loud (with the voice of the Arch Angel and the Trumpet of God). It’s going to be very visible (every eye will see Him). And it won’t be a secret. The timing of the event will be like a thief in the night, because most won’t be expecting it. But not the event itself.
Below are some scriptures which show why the Rapture teaching is not biblical.
For then shall be great tribulation, such as was not since the beginning of the world to this time, no, nor ever shall be.
Matthew 24:21,22
And except those days should be shortened, there should no flesh be saved: but for the elect’s sake those days shall be shortened.
If the elect were going to be taken to heaven beforehand, why would the days need to be shortened for their sake?
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:
1 Thessalonians 4:16,17
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.
At the second coming, when Jesus returns for us, the dead in Christ shall rise first. Then, those of us alive at the time and still remaining on the Earth, will be caught up, together with those just raised back to life, into the clouds to meet the Lord in the air. We weren’t already with Him in the air, it happens here at the very end.