Praise be to Allaah.
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 the result of its previous actions, and thus it moves from one body to another. This is one of the falsest of false beliefs, and one of the worst forms of kufr or disbelief in Allaah, His Books and His Messengers. For belief in the Hereafter, the Reckoning, Paradise and Hell are 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.
The Islamic understanding of the Resurrection is stated clearly in the Book of Allaah and the Sunnah of His Messenger (peace and blessings of Allaah be upon him). For example, Allaah says (interpretation of the meaning]:
“Everyone shall taste death. Then unto Us you shall be returned”
[al-‘Ankaboot 29:57]
“To Him is the return of all of you. The Promise of Allaah 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”
[Yoonus 10:4]
“The Day We shall gather the Muttaqoon (the pious) unto the Most Gracious (Allaah), like a delegation (presented before a king for honour).
And We shall drive the Mujrimoon (polytheists, sinners, criminals, disbelievers in the Oneness of Allaah) 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 everyone of them will come to Him alone on the Day of Resurrection (without any helper, or protector or defender)”
[Maryam 19:94-95]
“Allaah! Laa ilaaha 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 Allaah”
[al-Taghaabun 64:7]
And there are other clear and unambiguous verses.
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 Allaah 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 Ibraaheem…” (Narrated by al-Bukhaari, 3100; Muslim, 5104).
And the Prophet (peace and blessings of Allaah 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 Allaah?” He said, “The tailbone.” (Narrated by Muslim, 5255).