Takfeer (تكفير) is an Islamic term for excommunication. Even if a Muslim insists that he believes in Islam, he can be declared a non-Muslim by other Muslims. When he is declared a non-Muslim, he is considered an apostate. The punishment for apostasy in Islam is the death penalty. In Islam Muslims shouldn't kill each other, however with takfeer, it's practically possible (although in their view they're killing non-Muslim apostates).
The word is from the root كفر (k-f-r), just like the word kaafir (كافر). Arabic has 10 basic verb forms. The word kaafir is from the form I and the word takfeer is from the form II. The form II is usually causative of form I. So takfeer means "cause to be a kafir" or "make him a kafir".
In the Quran
The word takfeer itself doesn't appear in the Quran , but there are many verses which declare believers to be kafirs.
For example Christians are declared kafirs:
They have certainly disbelieved (كفر) who say that Allah is Christ, the son of Mary
Christians consider themselves believers, but the Quran declared them as kafirs.
A more general takfeer can be found at the end of this verse:
Indeed, We sent down the Torah, in which was guidance and light. The prophets who submitted [to Allah ] judged by it for the Jews, as did the rabbis and scholars by that with which they were entrusted of the Scripture of Allah, and they were witnesses thereto. So do not fear the people but fear Me, and do not exchange My verses for a small price. And whoever does not judge by what Allah has revealed - then it is those who are the disbelievers (الكافرون).
If someone doesn't judge by what Allah has revealed, then he is a kafir. But what Allah revealed can be understood differently by different Muslims. Every Muslim sect thinks that what they believe is what Allah revealed. And therefore other Muslim sects who have a different understanding have a wrong understanding and they don't judge by what Allah has revealed.
The death penalty
The concept of takfeer wouldn't be so dangerous if it wasn't combined with the death penalty for apostasy:
Narrated 'Ikrimah: That 'Ali burnt some people who apostasized from Islam. This news reached Ibn 'Abbas, so he said: "If it were me I would have killed them according to the statement of Messenger of Allah (ﷺ). The Messenger of Allah (ﷺ) said: 'Whoever changes his religion then kill him.' And I would not have burned them because the Messenger of Allah (ﷺ) said: 'Do not punish with the punishment of Allah.' So this reached 'Ali, and he said: "Ibn 'Abbas has told the truth."
- The caliph Ali who burned people alive is considered one of the "rightly guided caliphs".
Takfeer without reason
It's forbidden to declare someone a Kafir without a reason. If a Muslim does takfeer without a reason, then he himself becomes a kafir:
Narrated Abu Huraira:
Allah's Messenger (ﷺ) said, "If a man says to his brother, O Kafir (disbeliever)!' Then surely one of them is such (i.e., a Kafir). "
It's from the chapter "Whoever calls his brother a Kafir without any grounds". So it applies only on takfeer without a reason.
Ten nullifiers of Islam
The concept of ten nullifiers of Islam was declared by Muhammad ibn 'Abd al-Wahhaab. The list is based on primary Islamic sources. According to islamqa.info there is a consensus of scholars on this issue, so we cannot dismiss it as "wahhabism":
The things that nullify Islam are ten; they were mentioned by Imam Muhammad ibn ‘Abd al-Wahhaab, and there is scholarly consensus on all of them except one, namely sihr (magic, witchcraft), which is subject to further discussion, but with regard to some forms of it, the scholars are unanimously agreed that it nullifies Islam, as we shall see below.
The ten nullifiers are:
- Shirk or associating others in worship with Allah
- Ascribing intermediaries to Allah
- Not regarding the mushrikeen as kuffaar (disbelievers)
- Believing that the guidance of someone other than the Prophet (blessings and peace of Allah be upon him) is more perfect than his guidance
- Hating something that was brought by the Messenger
- Ridiculing any aspect of the religion of the Messenger
- Sihr (magic or witchcraft)
- Supporting the mushrikeen and helping them against the Muslims
- Believing that it is permissible for some people to go beyond the bounds of sharee‘ah
- Turning away from the religion of Allah, and not learning it or acting in accordance with it