Ibadism is the oldest surviving branch of Islam and the one almost no general account mentions, and it is neither Sunni nor Shia.
It descends from the Kharijites, the party that broke with Ali in 657 after he agreed to arbitration at the battle of Siffin, on the principle that judgement belongs to God alone and that a leader who errs forfeits his office. Kharijite militancy became a byword in Islamic history for declaring other Muslims unbelievers and killing them, and this is where Ibadis part company with their own ancestry. The moderate wing that formed at Basra in the late seventh century around Abd Allah ibn Ibad and Jabir ibn Zayd rejected that violence, held that other Muslims who differed were not polytheists, and permitted ordinary dealings, intermarriage and coexistence with them.
Ibadis themselves generally reject the Kharijite label altogether, and modern scholarship is divided over how much continuity there really is between the quietist Basran school and the militant groups. Both accounts should be held: the historical descent is real, and the repudiation of Kharijite practice is also real and is very old.
Doctrinally the school is austere and rationalist in cast. Ibadis hold that the Quran is created rather than eternal, a position they share with the early Mutazila and which sets them directly against the Sunni doctrine of the uncreated word of God. They deny that the blessed will see God with their eyes in paradise, on the ground that vision would imply a body. On grave sin they take a hard line, holding that the unrepentant Muslim sinner is not an unbeliever but is also not a believer, and faces the fire.
Ritually and legally they are close to Sunni practice, with their own school of law and their own body of hadith.