Sufism is the item on this list that fits the category worst, and the misfit is informative. It is not a branch of Islam parallel to Sunni and Shia; it is an inward dimension found within both, and the great majority of historical Sufis were ordinary Sunnis of one or another school of law.
The name is usually derived from suf, wool, for the coarse garment of early ascetics. The aim is ihsan, to worship God as though you see him, and the working method is dhikr, the disciplined remembrance of God, most often the repeated invocation of the divine names, sometimes silent, sometimes chanted aloud, and in some orders joined to music, poetry, breath control or movement. What Sufism adds to ordinary practice is not a different law but a programme of the interior: the purification of the self, the naming and treatment of its diseases, and a mapped sequence of stations and states along the way.
The relationship to the law has been the tradition's central problem and its central defence. Sufis have generally insisted that the path is walked within the law and never around it, that the outward observance is the ground on which inward work is built. Some individuals pressed the experience of unity to formulations that scandalised: al-Hallaj was executed at Baghdad in 922 after declaring I am the Truth, and Ibn Arabi's teaching on the unity of being has been argued over for eight centuries.
Al-Ghazali, in the late eleventh century, is the figure who made Sufism respectable to the jurists, by demonstrating in the Revival of the Religious Sciences that the inward disciplines were the completion of the law rather than a rival to it. His success is why Sufism was mainstream Islam for most of the next eight hundred years.