Sufism's organisational unit is the tariqa, the order or path, and its authority runs through persons rather than institutions.
A seeker attaches to a shaykh or pir, a master, who prescribes the practice, monitors the interior life and gives permission to proceed. The master's own authorisation traces back through a documented silsila, a chain of transmission, usually reaching the Prophet through Ali or Abu Bakr. That chain is the credential, and it is what makes the tradition a transmission rather than a technique.
The orders formed from the twelfth century and are named for their founders, Qadiri after Abd al-Qadir al-Jilani, the most widespread; Naqshbandi, distinguished by silent dhikr and a strong emphasis on legal orthodoxy, dominant across Central Asia and Turkey; Chishti, whose shrines in India are visited by Muslims and Hindus alike and whose devotional music, qawwali, is one of the great living forms of Islamic art; Shadhili in North Africa; Mevlevi in Anatolia, founded around Rumi, whose whirling ceremony is the most photographed Sufi practice in the world and is a formal liturgy rather than a display.
Around the orders grew the institutions: the lodge, the shrine of the founding saint, the endowment supporting both, and the annual festival at the tomb. Shrine visitation is where Sufism meets popular religion, and it is what the modern reform movements most object to.
That objection is the tradition's current difficulty. Salafi and Wahhabi currents regard veneration at tombs and intercession through saints as association of partners with God, the gravest offence in Islam. In several countries this has moved from polemic to violence: Sufi shrines and gatherings have been bombed in Pakistan, Egypt, Iraq, Libya and Mali, and shrines destroyed outright where armed movements have taken territory.