Short answer
Sufism (tasawwuf) is the inner, spiritual dimension of Islam that emphasises purifying the heart, remembering God and good character. Many Sufi orders have been important in spreading Islam, but some practices of popular Sufism are criticised by other Muslims.
The fuller answer
Sufis follow the teachings of a spiritual guide in orders such as the Naqshbandi, Qadiri and Shadhili, and practise a lot of remembrance of God (dhikr), and some use poetry and music, as in the whirling of the Mevlevi order of Rumi. Rumi and Rabia al-Adawiyya are well known in the world for their writing on the love of God.
Many scholars through history were both jurists and Sufis, and see it as part of the religion of excellence (ihsan). Some Muslims, such as Salafis, reject practices they consider innovations, like asking the dead saints for help. Sufism is a living part of Muslim life in many countries, from Senegal to Indonesia.
This answer follows the mainstream Sunni understanding and is general learning, not a fatwa. Where Muslims differ, the answer says so. Have a question of your own? Write to us.