This school held that reason alone is sufficient to understand the nature of Allah ("God") and existence, and guide man's actions to right or wrong results, and to perceive the difference between morality and immorality. Prophecy and revelation were thus not absolutely essential to perceive the truth, and men in general were free to do as they wished.
They were in general supportive of early Greek philosophy and reacting to kalam, the traditional process of interrogation, that had dominated Muslim thought since its beginnings. Many Muslims saw them as defying the revelation of the Quran.
Most consider Al-Kindi (d. 866) the founder of the school. His encyclopedic output of 270 books laid the foundations for the later work of Al-Farabi (d. 950) who wrote on the state, for Ibn Sina (d. 1037) on medicine, and Ibn Rushd (d. 1198). All wrote commentary on Aristotle and are widely considered to have influenced Aquinas to a great degree.
The Mutazilites are widely considered to have extended ijtihad beyond religion into predecessors of what we today know as the scientific method. They were willing not only to quote but to correct Aristotle (unlike Aquinas), compare his work with that of others, and think critically on their own. This sets them apart from predecessors, contemporaries, and their immediate successors, both in the Muslim and Christian community.
The Abbasid Caliphs were generally supportive of their work, in part due to the additional latitude of action that was implied by their philosophy. A ruler who did not need to consult the religious authorities (ulama) was better able to respond to events, they thought, and indulge his whims. Perhaps for this reason, the Mutazilite school was very strongly opposed.
The Caliphs may also have realized that mastery of science implied mastery of technology and thus weaponry. Perhaps for this reason, the school was extremely influential, and was not truly extinguished until "The Incoherence of the Philosophers", by Al-Ghazali of the Asharite school, the fiercest opponents of the Mutazilites, became the dominant theory of Islamic thought.
The eloquent defense of rationalism by Ibn Rushd was the last gasp of the Mutazilite school. It included the notable line "to say that philosophers are incoherent is itself to make an incoherent statement", which sounds more or less like a modern view. It was, although still not a popular one.
By the dawn of the 13th century the Asharites were clearly in ascendance, due in part to the increasing diversity of Islam and the difficulty of maintaining a coherent discourse in the Arabic language that would reach all Muslims everywhere. The local ulama and tarika had a clear advantage in that they were trusted by their neighbors, even to translate texts into the non-Arabic languages. They gradually "closed the doors of ijtihad" and instituted taqlid ("imitation" or "precedent") as the way shariah judgements were justified in fiqh, leading to the schools of classical Muslim jurists. Mutazilites were forgotten, although their works were translated and some were later influential in the European Renaissance.
See also: Early Muslim philosophy, Asharite, Kalam, ijtihad