Sufism, or Tasawwuf, is a science whose objective is to repair the heart, and turn it away from everything except for Allah ﷻ, the Most High. This was a science present in the time of RasoolAllah 
, but it wasn’t given a title as such.
The idea of Sufism is to attain Ihsan (as mentioned in the famous Hadith of Islam, Imaan, and Ihsan. Ihsan is the rank where you worship Allah ﷻ as if you are seeing Him).
This goal isn’t one that is achieved easily, and requires strenuous effort. This striving to cleanse the heart is known as Tazkiya, and is mentioned in the Qur’an in Surah 91, verse 9:
“Successful is he who causes it [the soul] to become pure.”
This is achieved by acting upon one’s knowledge, being in constant remembrance, ridding the heart of ill qualities and being constantly aware of Allah ﷻ which results in true God-consciousness.
This is very difficult to complete alone. As a result, we need a Shaykh who can teach us how to become God-conscious. In the same way RasoolAllah had taught the companions how to perfect their actions, character and thoughts; a Shaykh has a similar function. Each Shaykh’s method of teaching this will vary depending on their Silsila (spiritual path).
A Silsila is a chain of transmission when it comes to Sufism. There are 41 Salaasil, each going through previous Shayukh eventually leading up to the Messenger . All of these orders trace their spiritual lineage through Sayyiduna `Ali 
, with the exception of one; it is only the Naqshbandi Silsila that traces its lineage through Sayyiduna Abu Bakr .
Tasawwuf is difficult to come across in its true essence, it didn’t necessarily carry a title in the time of RasoolAllah but it was put into practice and resulted in the formation of beautiful examples which we still benefit from today. Sayyiduna Abu Bakr’s honesty, Sayyiduna Umar’s strictness, Sayyiduna Uthman’s modesty, and Sayyiduna Ali’s prowess in knowledge and passing judgements are all prime examples of what Tasawwuf leads to; a perfection of character.
RasoolAllah says in a Hadith found in the collection of Imam Nawawi 
,
“Goodness is found in good character.”
And for us, Sufism provides a way to achieve good character. Putting our own desires aside, and rushing to fulfil the rights of those who are around us is a key aspect of Sufism, and sees the seeker focusing on their relationship with Allah ﷻ, the Most Exalted.
As a Naqshbandi, our journey as seekers will see us focus on certain aspects of Sufism. For example, this Tareeqah will emphasise on service of others, humbling of oneself and constant remembrance of Allah ﷻ. This allows for one to eventually put out the fire of arrogance that is found within the heart of man, and keeps one’s mind occupied with the fact that Allah ﷻ is a witness to all that we do, leading to one only performing those actions which gain His ﷻ, pleasure. May Allah ﷻ, the Most High, give us the best of abilities to do actions that are pleasing to Him ﷻ.