Reincarnation is the belief that when a person dies their spirit will be placed in a new born babies body. all current people have lived other lives before they were born this time around.  Some groups add Karma, and the transmigration of souls to this doctrine.

Resurrection is the Christian belief that after death the body will be recreated or brought back to life and the soul reunited with that body. The spirit remains without a body until this occurs.

In 1 Corinthians 15:12-18 Paul stated that there were those who did not believe in a resurrection. He stated that if there is no resurrection then Jesus was never raised, which makes him a false witness and our faith useless.
We have the testimony that Jesus was crucified, buried for three days, then resurrected in his physical body. The disciples recognized and touched him with their own hands. (John 20:26-27) Jesus said his resurrected body was was still flesh and bone, and He ate fish with his friends.
At Jesus accession the angel stated that Jesus would come back in like manner as he went, (Acts 1:11) a Physical body will descend from the heavens and his feet will touch the mount of olives. (Zach 14:4)
Martha stated she believed that Jesus would resurrect her bother Lazarus in the last days. (John11:24) But Jesus did it then. Lazarus was not reborn into a babies body, but physically came out of the tomb in his old body.
In 1 Thessalonians 4:16 Paul states there would be a resurrection of believers who have died, then the rapture of believers who are still alive all in a twinkling of a eye. Again not a reincarnation but a resurrection.
Paul went on to say that it is appointed for a man once to die and after that the judgment, not rebirth. (Hebrews 9:27)

In conclusion we can see that the bible does not confuse resurrection with reincarnation, and that all Christians believe in the resurrection of the dead. If there is reincarnation there is no resurrection. You can't wait and be raised in your old body and be born in a new on at the same time. As Paul put it if there is not resurrection (because there is reincarnation) then Jesus did not raise and Christianity is false.

Scriptures taken out of context:

1. The man born blind in John 9:2
   New Agers will argue that when the disciples asked if the man’s blindness was cause by his own sins or the sins of his parents, they considered the possibility of him sinning before he was born thus causing himself to be born blind. And thus a reincarnation.
   But Jesus says neither was the case, he was born that was for the glory of God.
2. John the Baptist come again as Elijah in Luke 1:17
   Likewise some try to say that John the Baptist was Elijah reincarnated, but scripture clearly says he came in the spirit and power of Elijah, not a reincarnation.

The Early Church Fathers on Reincarnation:

• Justin Martyr 165 AD, Dialogue 5 - The soul is created but does not die, but a physical death goes either to a better place or a worse place.
• Tatian 166 AD, Greeks 6 - We did not exist before we were born and we do not reincarnate. There will be a physical resurrection of the body.
• Athenagoras 177 AD, 36 - Starts a long discussion on the Physical resurrection of the Body. Christians do not believe the soul dissolves or in reincarnation.
• Irenaeus 178 AD, Against Heresies 2.29 - Christians believe in the resurrection of the Physical body.
  • Against Heresies 2.33 - Reincarnation is absurd.
  • Against Heresies 2.34 - Souls are only created once, are immortal, and can be recognized like Abraham and Lazarus. 
• Clement of Alexandria 192-202 AD, Stromata 3.14 - Souls do not pre-exist.
• Commodianus 240 AD, 26 - There is an eternal hell
• Tertullian 190-210 AD, Marcion 5.9&10 - Christians will physically resurrect.
  • Treatise of the Soul 1.27 - Soul and body conceived at the same time.
  • Treatise of the Soul 1.28-32 - The Pythagorean doctrine of transmigration is false.
  • Treatise of the Soul 1.58 - Souls do not sleep after death. at death all souls go to Hades.
• Minucius Felix, 35 - Unrighteous men have eternal punishment
• Origen 230 AD, 8:25 or 30? - Christians do not believe in reincarnation. 
  • John 6.7 - Reincarnation (Transcorporation) is a false doctrine.
  • Matthew 10.20 - “transmigration of souls into bodies” is a false dogma
  • Matthew 13.1 - John the Baptist was not the reincarnation of Elijah the prophet. There is not such thing as reincarnation.
• Lactantius 285 AD, Divine Institutes 3.18 - There is no reincarnation.