Sufism is an movement that has developed inside Islam. It is a collective name for groups who - besides common Islamic practice - have developed exercises an rules (Tariq) that mostly aim to feel/experience a tight mental contact with God.
Those exercises do not contradict the Quran or the teachings of Muhammad (p.b.u.h) but they don't originate there.
The Sufi tradition arose from early Islamic ascetic communities, which often attracted Christian ascetic schools, the members of which converted to Islam, either collectively or individually. Teachings and traditions in Sufism has parallels with the (relatively scarce) writings of those groups, and most of the Islamic traditions on Jesus (p.b.u.h) have been related by Sufi. Parallels to practices found in Asian religions may come from an indirect cultural exchange or be parallel findings based on the common properties of the human minds and souls.
Sufism has developed in Sunni Islam and most schools are Sunni. Nevertheless, there is also a Shia Sufi congregation, and a Alevite congregation.
The attitude towards Sufism among other scolars differs. For the Alevites, it's an important part of their religion.
Shia Imams have often been and still are often sceptical or hostile towards Sufism in Shia not so much because of the practice but because the Sufi schools have their own Imams in parallel to their own hierarchy.
Among Sunni scolars we find positive as well as sceptical attitudes toward Sufism, Salafi reject Sufism as they teach that nothing should be added to Islam that hasn't been founded by Muhammad (p.b.u.h) and his companions. Radical Salafists denounce Sufi as kafirun but main stream Islam as well as the Sufis themselves see in it a particular practice and tradition inside Islam.
There are also some modern groups in the U.S. who call themselves Sufi but only copy some practice without the religious background and faith. They are not Muslim, and Muslim congregations do not see them as real Sufi.