Exegesis. A system of interpretation of the Quran which follows a similar methodology as that used by the Rabbis in their interpretation of the Torah.
The five traditional sources for commentary of the Quran are:
This is regarded as the highest form of tafsir, on the belief that the Qu'ran is the word of Almighty God and authoritative when it explains itself. A related Muslim belief is that the Qur'an is c) free of contradiction, and that apparent inconsistencies in its message are inevitably resolved through closer study of the Qur'anic text.
Muslims believe that Muhammad was sent, among other reasons, to explain and clarify the Qur'an to people. The accounts of Muhammad's teaching recorded in the hadith collections thus contain much tafsir of the Qur'an, under titles such as "Meaning of Qur'anic verses." An authenticated hadith is regarded the second highest form of tafsir, because the Islamic prophet is explaining it -- but many of these traditions are disputed.
3The reports of the Sahaba.
The Sahaba, or companions of Muhammad, also interpreted and taught the Qur'an. If Qur'anic explication is absent, and there is no authentic tradition deriving from Muhammad, then a consensus of the companions may be helpful in interpreting a certain verse. Scholars have an obligation to follow that consensus.
4The reports of those who learned from the companions.
These people grew up in an environment with people who had known the Prophet, so their insight is the next in line of the sources of tafsir. (In addition, the recorded practice of those who lived in the Prophet's city of Medina carry special weight in the Maliki school.)
A qualified scholar's personal reasoning (deductive logic and personal evaluation of arguments) is the final method of understanding the Qur'an; it exists in conjunction with the other four. See Ijtihad. Early caliphs are strongly associated with this method of tafsir.