The Rapture is one of the events that is believed will occur during the end-times. There are several different end-times scenarios that are all based on slightly different interpretations of Scripture, but the three major ones all agree that it will occur.
(A comparison of some of the views can be found here.)
The term "Rapture" means "Caught up". The word itself comes from 1 Thessalonians 4:17 (KJV)
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.
The end times events are covered in many books in the Bible, but primarily in the Book of Daniel, The Book of the Revelation, 1 Thessalonians, and Matthew 24. Your friend's description is known as the "Pre-Tribulation Rapture" interpretation of end-times events, which is held to by many denominations, but is definitely not the only interpretation.
One minor point, is that you mentioned that the "good people" will be raptured. That's not quite right. The Christians will be Raptured according to Scripture. Christians are not Christians because they are good, or because they go to Church, or because they claim to be in a certain religion.
The Biblical definition of a Christian is someone who has been saved, or redeemed by Christ. Christianity isn't about being "good enough" to get to heaven, or about religious traditions, it's about Christ, and the free gift of eternal life that we gain by simply accepting His free gift, by faith. (More here.)
As far as predicting the date, that's a fool's game. Christ Himself said that nobody would know the date, and that it would take us by surprise.
Mark 13:28-37 (King James Version)
28 Now learn a parable of the fig tree; When her branch is yet tender,
and putteth forth leaves, ye know that summer is near:
29 So ye in like manner, when ye shall see these things come to pass,
know that it is nigh, even at the doors.
30 Verily I say unto you, that this generation shall not pass, till
all these things be done.
31 Heaven and earth shall pass away: but my words shall not pass away.
32 But of that day and that hour knoweth no man, no, not the angels
which are in heaven, neither the Son, but the Father.
33 Take ye heed, watch and pray: for ye know not when the time is.
34 For the Son of Man is as a man taking a far journey, who left his
house, and gave authority to his servants, and to every man his work,
and commanded the porter to watch.
35 Watch ye therefore: for ye know not when the master of the house
cometh, at even, or at midnight, or at the cockcrowing, or in the
morning:
36 Lest coming suddenly he find you sleeping.
37 And what I say unto you I say unto all, Watch.
The number of people that have claimed to know the date of the Rapture, and have seen that date come and go are countless. We don't know the dates. We can see the signs of the times - the types of events and state of the world that is predicted, but not down to the date.
It could be 2 seconds from now, it could be a hundred years from now, or it could be longer. We just don't know, but it is promised that it will occur, and that we are to watch for it.
So, if the Bible is true, and it is the Word of God as claimed, and Jesus was the Son of God, and The Way, The Truth, and The Life, and the only way to Heaven there's no time to lose. If you've not already accepted Christ as your Savior, now's the time.
