Takfir is the notion that an unbeliever, even a Muslim, may be excommunicated the moment that individual does not follow Sharia, or Islamic law, in its strictest sense. Merely fulfilling Islam's five pillars isn't enough, and may well be a cover for otherwise un-Islamic behavior if the individual is still susceptible to Western, Christian, Jewish, "Zionist" influences. Such susceptibility is equated with a return to jahiliyah, or the "state of ignorance" that prevailed before the advent of the Prophet Muhammad and Islam in the 7th century. Modern jahiliyah is considered even worse by believers in takfir, because Muslims today have the means to differentiate between true belief and ignorance. When they choose the latter, their crime is greater even than that of non-Muslims.
Excommunication for Muslims is the equivalent of apostasy--itself a justification for capital punishment in extremist interpretations of Islamic purity.
The notion of takfir has been used by al-Qaeda and similarly militant and terrorist Islamist organizations to justify the killing of innocent people, Muslims as well as non-Muslims.
But takfir is a controversial concept in Islam. Most Muslims, including most clerics and fundamentalist organizations such as the Muslim Brotherhood, reject the concept as un-Islamic, deviant and dangerously divisive, as it is more likely to tear apart the ummah, or Islamic community than bring it together.