Reincarnation in Islam
Question

I have a member of my family who believes in reincarnation which is a view I strongly oppose. What is the Muslim explanation behind this, as I wish to correct them (they have fallen from the faith and this may help)?
Praise be to Allah.What is reincarnation?
What is meant by reincarnation or the transmigration of souls is that when the body dies, the soul moves to another body, where it will be happy or miserable as a result of its previous actions, and thus it moves from one body to another. 
Ruling on reincarnation 
This is one of the falsest of false beliefs, and one of the worst forms of disbelief in Allah , His Books and His Messengers. For belief in the Hereafter, the Reckoning, Paradise and Hell is among the things that are well-known in the teachings of the messengers and in the words of the Books which were revealed to them. Belief in reincarnation is tantamount to disbelief in all of that. 
Resurrection in the Quran 
The Islamic understanding of the Resurrection is stated clearly in the Book of Allah and the Sunnah of His Messenger (peace and blessings of Allah be upon him). For example, Allah says (interpretation of the meaning]: 
“Everyone shall taste death. Then unto Us you shall be returned.” [al-'Ankabut 29:57]
“To Him is the return of all of you. The Promise of Allah is true. It is He Who begins the creation and then will repeat it, that He may reward with justice those who believed and did deeds of righteousness. But those who disbelieved will have a drink of boiling fluids and painful torment because they used to disbelieve.” [Yunus 10:4] 
“The Day We shall gather the Muttaqun (the pious) unto the Most Gracious (Allah), like a delegation (presented before a king for honour). And We shall drive the Mujrimun (polytheists, sinners, criminals, disbelievers in the Oneness of Allah) to Hell, in a thirsty state (like a thirsty herd driven down to water).” [Maryam 19:85-86]
“Verily, He knows each one of them, and has counted them a full counting.  And all of them will come to Him alone on the Day of Resurrection (without any helper, or protector or defender).” [Maryam 19:94-95]
“Allah! La ilaha illa Huwa (none has the right to be worshipped but He). Surely, He will gather you together on the Day of Resurrection about which there is no doubt.” [al-Nisa 4:87]
“The disbelievers pretend that they will never be resurrected (for the Account). Say (O Muhammad): Yes! By my Lord, you will certainly be resurrected, then you will be informed of (and recompensed for) what you did; and that is easy for Allah.” [al-Taghabun 64:7]
 And there are other clear and unambiguous verses.
Resurrection in the Sunnah
There are innumerable references in the Sunnah which mention and confirm the Resurrection, and give details concerning it. For example, the Prophet (peace and blessings of Allah be upon him) said: “You will be resurrected barefoot, naked and uncircumcised.” Then he recited (interpretation of the meaning): “As We began the first creation, We shall repeat it. (It is) a promise binding upon Us. Truly, We shall do it” [al-Anbiya 21:104]. (Then he said:) “And the first one to be clothed on the Day of Resurrection will be Ibrahim…” (Narrated by al-Bukhari, 3100; Muslim, 5104)
 And the Prophet (peace and blessings of Allah be upon him) said: “In man there is a bone which will not be consumed by the earth, and from it he will be regenerated on the Day of Resurrection.” They asked, “What bone is that, O Messenger of Allah?” He said, “The tailbone (coccyx).” (Narrated by Muslim, 5255)
And he said: “The sun will be brought close to mankind on the Day of Resurrection until it will be as close to them as a ‘mil’.” Salim ibn ‘Amir [one of the narrators of this hadith] said: “By Allah, I do not know whether by ‘mil’ he meant a distance on land (a mile) or the stick with which kohl is applied to the eyes. The Prophet (peace and blessings of Allah be upon him) said: “Then the people will be submerged in sweat according to their deeds; some will be in sweat up to their ankles, some up to their knees, some up to their groin and some up to their jaws.” He said: and the Messenger of Allah (peace and blessings of Allah be upon him) pointed to his mouth. (Narrated by Muslim, 5108)
 And he said: “I will come to the gate of Paradise on the Day of Resurrection and ask for it to be opened. The gatekeeper will ask, ‘Who are you?’ I will say: ‘Muhammad.’ He will say, ‘I was commanded not to open it for anyone before you.’” (Narrated by Muslim, 292)
 And there are other similar ahadith.
 The idea of the transmigration of souls is a rejection of these texts and a denial of the Resurrection.
Does the soul move to another body?
The Islamic teachings concerning torment or blessing in the grave and the questioning of the two angels clearly prove that the soul of man does not move to another body, rather it and the body experience torment or blessing, until mankind will be gathered to their Lord.
Imam Ibn Hazm (may Allah have mercy on him) said: 
“It is sufficient refutation of their idea to note that there is consensus among all Muslims that they are disbeleivers and that those who believe in this idea are following something other than Islam, and that the Prophet (peace and blessings of Allah be upon him) taught something other than this.” (al-Fisal fi’l-Milal wa’l-Ahwa wa’l-Nihal, 1/166)
The belief that the body will perish and will not be restored to experience blessing or punishment is a means that leads man to indulge in desires, wrongdoing and evil. This is what the Shaytan wants for those who follow this corrupt belief, in addition to pushing them further into kufr when he makes them believe in this false idea.
 What you must do is to advise this person and remind him of the words of Allah and the words of His Messenger (peace and blessings of Allah be upon him), and call him to repent from this disbelief. If he repents and turns to Allah, all well and good, otherwise keep away from him and warn others against listening to him, and tell people that you reject his beliefs, lest people be deceived by him.
 And Allah knows best.