As dleyva3 mentioned, the term comes from 1 Thessalonians 4:17. Our word "rapture" comes from the Latin word for "caught up".
The doctrine that faithful Christians would be whisked away to be spared tribulation is a relatively new interpretation of that verse. The traditional Christian belief was that Jesus would return once, to bring judgment. At his return, the faithful would be so caught up with joy that they would rise into the air to accompany him in his descent.
The fourth century preacher John Chrysostom put it this way:
If He is about to descend, on what account shall we be caught up? For the sake of honor. For when a king drives into a city, those who are in honor go out to meet him; but the condemned await the judge within. And upon the coming of an affectionate father, his children indeed, and those who are worthy to be his children, are taken out in a chariot, that they may see and kiss him; but those of the domestics who have offended remain within. We are carried upon the chariot of our Father. For He received Him up in the clouds, and “we shall be caught up in the clouds.” (Acts i. 9.) Seest thou how great is the honor? and as He descends, we go forth to meet Him, and, what is more blessed than all, so we shall be with Him. —Homily #8, 1 Thessalonians 4:15-17
A few Christians may have had other ideas through the centuries, but the first one to really popularize the idea that Jesus would return twice—first to take the faithful away, and then later for judgment day—was John Nelson Darby in the early 19th century. That's the view that is commonly known as "the rapture" today.