Tasawwuf is a concept in Islam, defined by scholars as the inner, mystical dimension of Islam. Tasawwuf is the soul of Islam. Its function is to purify the heart from the lowly bestial attributes of lust, calamities of the tongue, anger, malice, jealousy, love of the world, love of fame, niggardliness, greed, ostentation, vanity, deception, etc. At the same time it aims at the adornment of the heart with the lofty attributes of repentance, perseverance, gratefulness, fear of Allah, hope, abstention, tauheed, trust, love, sincerity, truth, contemplation, etc.
As for the origin of the word Tasawwuf, it may well be from Sufi, the person who does Tasawwuf, Lexicographers have identified a number of source words from which the term Sufi is derived. The most widely accepted word from which Sufi is derived is “suf” meaning wool. The earlier ascetics often donned woollen garments to express their inner detachment from the world and their rejection of the excessive materialism of the earlier Islamic dynasties.
Other terms that suggest themselves as source words are the following:
Safa, meaning purification.
Safwe, which means those who are selected.
Suffa, meaning a bench or low veranda. During the time of the Prophet (may the peach and blessings of Allah be upon him) a number of Companions disengaged themselves from normal worldly activities and devoted themselves to an ascetic way of life. They came to be known as the Ashaab us-Suffa or “Companions of the Bench”. They spent the greater part of their lives in acts of devotion on a low veranda in the vicinity of Masjid e Nawwabi (The mosque of Hazrat Muhammad (SM).
Saff, meaning rank, line, or row. The first row in congregational prayers in Islam has been accorded a special status for it symbolises those who are in the first rank of spirituality.

The term “Sufi” could be traced dated back to the time of “Tabeyin” as Hazrat Hasan al-Basri who died 110 years after the Hijra, is reported to have said, “I saw a Sufi circumambulating the Kaaba, and offered him a dirham, but he would not accept it.” It therefore seems better to understand Tasawwuf by first asking what a Sufi is; and perhaps the best definition of both the Sufi and his way, certainly one of the most frequently quoted by masters of the discipline, is from the sunna of the Prophet (Allah bless him and give him peace) who said:
Allah Most High says: “He who is hostile to a friend of Mine I declare war against. My slave approaches Me with nothing more beloved to Me than what I have made obligatory upon him, and My slave keeps drawing nearer to Me with voluntary works until I love him. And when I love him, I am his hearing with which he hears, his sight with which he sees, his hand with which he seizes, and his foot with which he walks. If he asks me, I will surely give to him, and if he seeks refuge in Me, I will surely protect him” (Fath al-Bari, 11.340–41, hadith 6502);
This hadith was related by Imam Bukhari, Ahmad ibn Hanbal, al-Bayhaqi, and others with multiple contiguous chains of transmission, and is sahih.
Some scholars opine that the word “Sufi” in Arabic emerged from the Greek word “Philasoof”. The first part “Phila” stands for lover and the other part “Soof” means wisdom and the full word “Philasoof” means the lover of wisdom and when the word “Philasoof” was Arabicised it became ‘Sufi”. But this concept or ideology of scholars like Al-Bairuni (973 AH-1048 AH) is not acceptable in its entirely, for the simple reason that the work of transferring the Greek Cyclopedia into Arabic language was undertaken only in approximately the half of the third century A.H. whereas since the word Sufi was already in vogue as part of the spoken Arabic language much earlier than that, the concept of the word Sufi as a derivation of Greek Philosophy does not remain tenable. The person who nicknamed ‘Sufi’ for the first time in the Arabic History was Abul Hashim Alkufi who died in the year 150 A.H. that is one hundred year before the work of translating Greek Literature into Arabic Language actually started.
The origin of the way of the Sufi thus lies in the prophetic sunna. The sincerity to Allah that it entails was the rule among the earliest Muslims, to whom this was simply a state of being without a name, while it only became a distinct discipline when the majority of the Community had drifted away and changed from this state. Muslims of subsequent generations required systematic effort to attain it, and it was because of the change in the Islamic environment after the earliest generations, that a discipline by the name of Tasawwuf came to exist. 
But if this is true of origins, the more significant question is: How central is Tasawwuf to the religion, and: Where does it fit into Islam as a whole? Perhaps the best answer is the hadith of Muslim Sharif, that Ameer-ul-Momenin Hazrat Umar ibn al-Khattab (Ra:) said: 

“As we sat one day with the Messenger of Allah (Allah bless him and give him peace), a man in pure white clothing and jet black hair came to us, without a trace of travelling upon him, though none of us knew him.
He sat down before the Prophet (Allah bless him and give him peace) bracing his knees against his, resting his hands on his legs, and said: “Muhammad, tell me about Islam.” The Messenger of Allah (Allah bless him and give him peace) said: “Islam is to testify that there is no god but Allah and that Muhammad is the Messenger of Allah, and to perform the prayer, give zakat, fast in Ramadan, and perform the pilgrimage to the House if you can find a way.”
He said: “You have spoken the truth,” and we were surprised that he should ask and then confirm the answer. Then he said:
“Tell me about true faith (iman),” and the Prophet (Allah bless him and give him peace) answered: “It is to believe in Allah, His angels, His inspired Books, His messengers, the Last Day, and in destiny, its good and evil.”
“You have spoken the truth,” he said, “Now tell me about the perfection of faith (ihsan),” and the Prophet (Allah bless him and give him peace) answered: “It is to worship Allah as if you see Him, and if you see Him not, He nevertheless sees you.”
The hadith continues to where ‘Umar said:
Then the visitor left. I waited a long while, and the Prophet (Allah bless him and give him peace) said to me, “Do you know, ‘Umar, who was the questioner?” and I replied, “Allah and His messenger know best.” He said, 
“It was Gabriel, who came to you to teach you your religion” (Sahih Muslim). 
This is a sahih hadith, described by Imam Nawawi as one of the hadiths upon which the Islamic religion turns. The use of din in the last words of it, Atakum yu‘allimukum dinakum, “came to you to teach you your religion” entails that the religion of Islam is composed of the three fundamentals mentioned in the hadith: Islam, or external compliance with what Allah asks of us; Iman, or the belief in the unseen that the prophets have informed us of; and Ihsan, or to worship Allah as though one sees Him. The Qur’an says, in Surat Maryam, 
“Surely We have revealed the Remembrance, and surely We shall preserve it” (Qur’an 15:9), 
and if we reflect how Allah, in His wisdom, has accomplished this, we see that it is by human beings, the traditional scholars He has sent at each level of the religion. The level of Islam has been preserved and conveyed to us by the Imams of Shari‘a or ‘Sacred Law’ and its ancillary disciplines; the level of Iman, by the Imams of ‘Aqida or ‘tenets of faith’; and the level of Ihsan, “to worship Allah as though you see Him,” by the Imams of Tasawwuf. 
The statements of four Imams of Shariat (Mazhab) on Tasawwuf

Imam Abu Hanifa
(85 H. – 150 H) – “If it were not for two years, I would have perished.” He said,
“for two years I accompanied Sayyidina Ja’far as-Sadiq and I acquired the spiritual knowledge that made me a Gnostic in the Way.” [Ad-Durr al-Mukhtar, vol 1]

Imam Malik
(95 H. – 179 H.) – “whoever studies Jurisprudence [tafaqaha] and didn’t study Sufism
[tasawwaf] will be corrupted; and whoever studied Sufism and didn’t study Jurisprudence will become a heretic; and whoever combined both will reach the Truth.” [‘Ali al-Adawi , vol. 2]
Imam Shafi’i
(150 – 205 AH.) – “I accompanied the Sufi people and I received from them three

knowledges: …how to speak; how to treat people with leniency and a soft heart… and they… guided me in the ways of Sufism.” [Kashf al-Khafa, ‘Ajluni, vol. 1]

Imam Ahmad bin Hanbal
(164 – 241 AH.) – “O my son, you have to sit with the People of Sufism, because they are like a fountain of knowledge and they keep the Remembrance of Allah in their hearts. they are the ascetics and they have the most spiritual power.” [Tanwir al-Qulub]

The above mentioned statements of the Imams of four mazhabs indicate clearly that they were the followers of tasawwuf and this pactice of tasawwuf has guided them to understand and practice the true assence of Islam. Now let us focus on how the world renouned sufi masters explained Tasawwuf:
Hazrat Junaid Bagdadi (Ra:) : Tasawwuf is that (extra sensori) perception in which Almighty Allah annahilates the self of you (outward existence) and grants you a fresh (revived) life to accompany His own Divine Self.
Hujjatul Islam Imam Gazzali (Ra:): The first step in the pathway of the Destination is that he (Salik) should ‘endeavour (hard work by way of Mujahidah); destroy and get rid of evil deeds, snap up all the extraneous relations and associations, and bend all his energy towards concentration of Divine Manifestation mirroring the Entity (Zaat) of Almighty Allah. When the traveller (Salik) attains this noble and sacred perfection, then Almighty Allah Himself becomes the Guardian and Protector of the heart of His servant and becomes the Guarantor of providing Light and Guidance from the Divine Treasures of enlightment inspiration. (Ihya-ul-Uloom)
Hazrat Ibne Sina (Ra:) : The man who is devoted to absolute abstinence of worldly pleasures is called ‘Zahid’ and the man who keeps almost every moments of his life devoted to worship is remembered among the people with the epithet of ‘Aabid’the man who keeps his attention perennially focused towards the Holy Omnipotence (Qudoos-e-Jabroot) and expects every moment the Manifestations of Divine Light in his inner self to illumine every bit of spectrum of his life as a whole, is recognized and venerated with the ever shining title of ‘Arif’ (Recognizant of Divine Manifestation in all the things in the universe) and the opinion of Hakeem Ibne Sina only such an exceptional individual among the countless creatures of Almighty Allah is worthy of being honoured and remembered with the tilte of a (true) Sufi. (AL-ISHARAAT)
Hazrat Mawlana Jalaluddin Rumi (Ra:) : The Sufi is drunk without wine; he is lost in his Master’s love. He is sated without food; shows opulence in poverty;joy in sorrow and loves even his enemies.   
Hazrat Gausul Azam Mir Mohiuddin Abdul Qader Gilan (Ra): “The Sufi is someone who makes that which the Lord of Truth wishes from him the object of his own wish.  He renounces this world, so it serves him, and his allotted shares [aqsam] coincide with his needs.  He achieves his purpose in this world, before the Hereafter, for his well-being is ensured by his Lord.
Hazrat Khawaja Garib Nawaz Moinuddin Chishti Ajmeeri (Ra:):To sit in the company of pious people is better than doing good and to sit in the company of evil (immoral) people is worse than doing evil
Imam Mujaddid Alf-Thani Ahmad Sirhindi (Ra:):Man’s own nafs is the most adamant obstructive curtain between man and Allâhu ta’âlâ. “Abandon thy nafs, and come to Me! Thy very self is the cloud hiding the sun thou art after! Know thyself,” says the divine Word. Pushing the nafs away from between requires a conscientious and delectable process. It cannot be described by words and writings. Nor is it something that can be learned by perusal. It has to be a gift that one was endowed with in the eternal past, and it has to be primed by the attraction of Allâhu ta’âlâ. Since we live in a world of causations, a Wali’s lecture will suffice, with the proviso that you should love the Walî. The more you love him, the more will you receive of the fayz and ma’rifats radiating from his heart, attaining perfection at the end. The hadith-i-sherîf which reads, “A person will be together with his beloved one,” expresses this fact. ”
(Second Volume, 29th Letter, Maktubat of Ahmad Sirhindi)
I would now like to conclude by quoting from the world renowned book of Tasawwuf “Kashf-ul Mahjoob” written by Hazrat Data Ganj Baksh Al- Hujwiri (Ra:) regarding his view on the origin of Tasawwuf and the list of the holy names of the Sufis amongst the Sahabahs. Hazrat Data Ganj Baksh Al-Hujwiri (Ra:) writes,” Known that all Muslims are agreed that the Apostle had a number of Companions, who abode in his Mosque and engaged in devotion, renouncing the world and refusing to seek a livelihood. God reproached the Apostle on their account and said: “Do not drive away those who call unto their Lord at morn and eve, desiring His face (Quran vi,52). Their merits are proclaimed by the Book of God, and in many traditions of the Apostle which have come down to us. It is related by Ibn Abbas that the Apostle passed by the people of the veranda and saw their poverty and their self mortification and said: “Rejoice! For whoever of my community preserves in the state in which ye are, and is satisfied with this condition, he shall be one of my comrades in paradise.” Among the Ahl-i Suffah were Bilal b. Rabah, Salman al-Farisi, Abu Ubayda b. al-Jarrah, Abu I-Yaqzan, Ammar b. Yasir, Abdallah b. Mas’ud al- Hudhali, his brother Utba b. Mas’ud Miqdad b. al-Aswad, Khabbab b. al- Aratt, Suhayb b. Sinan, Utba b. Ghazwan, Zayd b. al- Khattab, brother of Caliph Umar; Abu Kabsha, the Apostle’s client; Abu I- Marthad Kinana b. al- Husayn al- Adawi, Salim, client of Hudhayfa al- Yamani; Ukkasha b. Mihsan; Mas’ud b. Rabi al- Faris; Abu Dharr Jundab b. Junada al- Ghifari; Abdallah b. Umar; Safwan b. Bayda; Abu Darda Uwaym b. Amir; Abu Lubab b. Abd al- Mundhir; and Abdallah b. Badr al- Juhani.”