The Rapture and all its other happenings on a Christian standpoint is suppose to happen without anyone really knowing whats going on. Rephrasing... life will continue to go on like normal and people will be completely unaware that its actually taking place. There are "signs" and other stuff that people may recognize but can otherwise be trivial. For myself, Biblically speaking, the Anti-Christ will be some world leader or a very well known person that is mortally wounded. (I could go into a bit more detail but I'll hold it for another post)
Even in Christian circles, the Rapture itself is in great debate of the "When" it'll take place because the Bible doesn't exactly stipulate but everyone is entitled their opinion about when it'll happen. Of course, the best deal is if it is Pre-Tribulation so that they don't have to go through all that stuff... but again, there are those that oppose that and say its during or even after Tribulation. Again, its still in debate.
As far as the term itself, Rapture was coined after a few verses found in the New Testament. Namely 2 Cor. 12:2, 4 and 1 Thess 4:16-17. The phrase is termed from "caught up" or "catching away" stated by the Apostle Paul within context. Since the bible was not written originally in English, the termed was used in place of its Greek translation "harpazo."