Probably Sufism http://www.ias.org/articles/Introduction_to_Sufism.html

Sufi. The Sufi are the mystics of Islam. They follow an ascetic lifestyle in search of the truth and some believe in the possibility of obtaining a mystical union with God through ritual dancing, music, or meditation. Sufis believe that Allah reveals himself to individuals on a personal level. There were many female mystics of Islam including Rabi'a of Basra. Little is known for sure about Rabi'a al -'Adawiyya al-Qaysiyya (known as Rabi'a of Basra), revered as one of the earliest and greatest Sufi mystic ascetics in Islam.

NO branch of islam embraces mystics. They're haram.

There is no branch that embraces mystics,and in fact they outlaw it. Like any religion ,the power is in it's teachings, and not the individual. If we were to communicate directly to god like the mystics do,religion would not be necessary.Sufism has been around for thousands of years, and started in ancient Arabia. It follows no form of religion,but is a spiritual practice,with meditation. It is mistakenly associated with Islam because it was founded geographically in that area.

In Response to earlier statements: Islam never banned mystecism as much of Islamic history is based upon mystecism. Mystecism is just a word and its meaning changes depending on the location and concept of the speaker. Mytecism in Islam existed from the very beginning and it was never haram or outlawed by the Prophet Muhammad PBUH but what was outlawed were the pre islamic Arabian polytheist practices that incorporated certain mystecism of other cultures and religions as well. Mystecism in Islam is centered in the Quranic reading and certain forms of worship. It is not known by the one who never learned it as it is only passed down from teacher to student. Don't get caught up in sterotypes of misunderstanding! there was a mystical tradition in Islam during the life of the Prophet Muhammad (Pbuh- peace be upon him) that has always existed same as there was a mystical tradition for Moses and for Jesus during their life times also. The orientalist term is sufism for the arabic term Tasawaf. The Term Sufi is a term taken from similar sounding Arabic words to indicate the piety and purity of those who practised the Islamic mystecism. The first four Islamic Teachers of law were all mystics as well as they were all fond of the family of the prophet Muhammad pbuh. One of them stated- may God Forbid if a student of religion were practising the islamic law without the sprititual path (tasawaf/tariqa/mystecism) as he would be corrupted! while the other student that practised the mystecism without the law (sharia) would be misguided! and the one who joined and practised the two would have the correct guidance! This is the Islam and what we refer to as Sufism. Orientalist have chose to to identify it Islamic mystecism. Concerning sects: The two major sects being Sunni and Shiya about which so much is in common but a minor difference about political matters concerning the family of the Propeht Muhammad peace be upon him (PBUH). It is fair to say that sunnis adore the family of the prophet as much as shiya but some shiya are extreme in this approach to the extent that they are against the way of the prophet Muhammad pbuh while some sunnis as you know are extreme while they are quick to consider this one and that one not really a muslim and even begin to hate the family of the prophet Muhammad pbuh something that is clearly against his teachings!!! As the prophet PBUH used to take the middle path that is the best way to proceed. The sufis or mystics can be found in both sects sunni and shiya and there are also those who say they are mystics and say they are sufis but have practices and traditions that is totally alien to Islam also. But no need to discuss that as the question is concerned with sects within Islam itself. It is well documented within islamic tradition and teaching that Extremism and extremist will perish! That can be applied to those who call them selfs strugglers in the path, Jurist of the law or mystics the same. The True Islamic path is the middle path. Any one who seeks to connect with Sufi orders on the net or otherwhise can find more rewarding and deeper information on this topic. Just a note as it is well documented within Islamic Belief that the muslims are to be devided in to 73 groups and only one of them is safe from the fire of Hell. No name was given so none of uf have the right to insiste who that one is or who the others are (72). There is much to be considered for those who judge others more than themselves and those who condemm others wrong but not their own. Lets all be humble in our approach to God and to humanity it is just a matter of time before GOD will settle for us our disputes beyond all doubts. In the meantime peace is what we should strive for as it states in Quran that those in paradise will greet each other with peace. Better to do that now in this life and spend our time to uplift one another peacefully to happiness and fairness rather than injustice and criminal activities expecting that we will merely be saved because we call our selves submitted( Muslims) or believers (Mumins) but we are not really Good doers (muhsineen) If you are looking for a way to find deeper realities to find God within ones self and self purification then Islamic Mystecism is definately a good way to proceed, but if you are looking for Atheism devil worship or witchcraft maybe you will not be satisfied with Islamic Mystecism as it is based upon worship and humility and brotherhood/sisterhood. Peace and best wishes