The term "Muʿtazila" refers to an influential Islamic theological school that had its heyday in the late eighth to tenth centuries. According to the standard accounts, the Muʿtazila first began as a splinter group of another early Islamic theological school, the Qadariyya. In the year 740 CE, the Basran scholar Wāṣil b. ʿAṭāʾ (d. 749) convinced ʿAmr b. ʿUbayd (d. 761) that the Qadarī al-Ḥasan al-Baṣrī was wrong in teaching that a "grave sinner" is an unbeliever. Wāṣil b. ʿAṭāʾ and some of his associates became known as the Muʿtazila. Before the Zanj Revolt (869-83) which devastated the city and its hinterland, Basra was a major center of intellectual and economic activity in the Caliphate.

Though the Muʿtazila disagreed with the Qadariyya on whether a grave sinner could be considered a Muslim, they agreed with the most well-known theological view that the Qadariyya propounded, namely that humans have free will and power (qadar) over their actions. This was in opposition to those Muslims who claimed that all was the work of God and He determines human action. Muʿtazilī theology was developed over the decades and became much more elaborated and complex than its Qadarī predecessor. The Muʿtazila also had a skeptical view of the hadith compared to some of their contemporaries. They only accepted hadith that were so widely transmitted (by twelve, twenty, or even seventy witnesses in each generation) that the tradition could not be doubted (mutawātir). Anthropomorphism [of God] (tashbīh) was vehemently rejected by Muʿtazila and they pejoratively called literalist Hadith scholars "mushabbiha" (anthropomorphists).

Perhaps the most famous (or infamous, depending on your point of view) Muʿtazilī doctrine was the khalq al-Qurʾān, the "createdness of the Quran" as opposed to the Quran being co-eternal with God. No doubt, no one contributed to the notoriety of this doctrine more than the Abbasid Caliph ʿAbd Allāh al-Maʾmūn (r. 813-33) who during the last year of his reign enforced the miḥna (inquisition) in which Muslim scholars (ʿulamāʾ) were pressured and sometimes forced to assert that the Quran was created. The most well-known victim of the miḥna was the proto-Sunni Hadith-scholar Aḥmad b. Ḥanbal (d. 855). The miḥna reached its height under the reign of al-Maʾmūn's nephew al-Wāthiq (r. 842-7) and ended in the first year of the reign of al-Mutawakkil (r. 847-61), who in contradiction to his immediate predecessors affirmed the symbiotic relationship between the Abbasid caliphs and the Sunnis.

Sources:

Blankinship, "The Early Creed" in The Cambridge Companion to Classical Islamic Theology

Cooperson, Classical Arabic Biography

Griffel, "Mu'tazilites" in Medieval Islamic Civilization, An Encyclopedia

Melchert, Ahmad Ibn Hanbal