This wonderful event will take place at the return of Jesus, otherwise known as His "Second Coming" (remember, His first coming was as a baby?).
1 Thessalonians 4:16 "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".
When Jesus "shouts", all the "dead in Christ" will be resurrected to life. They will not be sick or hurting in any way when they come back to life but will be raised with a new body and at that time they will be given immortality (meaning that they will never die again). These "rebuilt" people will still be recognizable to everyone who knew them. (1 Corinthians 13:12 "For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known.")
In the Bible, 1 Timothy 6:16 speaks of God as being the only One who "hath immortality". This means that those who have died, even Christians, died as all humans. All humans are mortals, we do not live forever unless Jesus gives us immortality. When does the Bible say that Jesus gives "immortality"? At the "last trump", when the "dead shall be raised incorruptible" and "put on immortality".
1 Corinthians 15:51-53 Behold, I show you a mystery; We shall not all sleep, but we shall all be changed, in a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed. For this corruptible must put on incorruption, and this mortal [must] put on immortality.