Your Christian friends are right; the Bible does not teach reincarnation (the idea that after death we come back to life on this earth).
In fact, the Bible explicitly rejects reincarnation, and teaches instead that we only die once, and then we enter eternity—either in heaven with God, or separated from Him in the place the Bible calls hell. The Bible says that "man is destined to die once, and after that to face judgment" (Hebrews 9:27). The passage you mention (which you can find in Matthew 11:7-14) does not teach reincarnation; Jesus is referring here to Elijah only as a symbol of John the Baptist, who announced Jesus' coming.
I have often asked myself why some people find the idea of reincarnation so attractive. One reason, I've found, is because they fear death, and they think reincarnation takes away that fear. Others realize they need to live a better life, and they hope they'll have another chance to do that.
If we know Christ, however, we don't need to fear death, nor do we have to worry about living a more perfect life. Christ came to take away our sins, and by His death and resurrection He opened heaven's door for us. Put your faith and trust in Him, for "Salvation is found in no one else, for there is no other name under heaven given to men by which we must be saved" (Acts 4:12).