Wikipedia defines Reincarnation as:

Reincarnation is the religious or philosophical concept that the soul or spirit, after biological death, begins a new life in a new body that may be human, animal or spiritual depending on the moral quality of the previous life's actions.

One of the primary beliefs of Islam is the belief in the Hereafter, resurrection, the Paradise and the Hell.

Allah says (interpretation of the meaning]:

“Everyone shall taste death. Then unto Us you shall be returned”[ al-‘Ankaboot 29:57 ]

And He says (interpretation of meaning):

“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 ]

He also says [interpretation of the meaning]:

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 ]

And He says (interpretation of the meaning):

“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 ]

And He says concerning the disbelievers who think they will not be resurrected (interpretation of the meaning):

“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-Taghaabun 64:7 ]

Allah says refuting those who claim reincarnation (interpretation of the meaning):

Do they not see how many of the generations We have destroyed before them Verily, they will not return to them. [ Ya-Sin 36:31 ]

Which means “do you not learn a lesson from those nations whom Allah destroyed before you of those who disbelieved in the Messengers. They came to this world only once, and will not return to it."

There are several hadeeth as well which mention and confirm resurrection. For example:

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.” (Narrated by Muslim, 5255).

The idea of reincarnation is rejection and denial of resurrection and hereafter and it has been refuted by texts of Islam.