What is the greater jihad?

Striving against one's own desires in the obedience of Allah. The perfect mujahid is the one who struggles with his desires.

Striving against one's desires is referred to as the greater jihad, or the greater battle, in Tasawwuf, and the term is established from the hadith in which the Prophet ﷺ said, “A Mujahid is the one who strives against his desires to obey Allah.” [Bayhaqi, Shu'ab al-Iman 1123]

The meaning is that the perfect mujahid, the warrior, is the one who struggles with his desires. Nafs e Ammara can be subdued by constant rebuke and rigor.

Imam Ghazali rahimahullah describes the process: “The first step in this path is that the seeker should undertake ‘mujahida’ or rigor, destroy and get rid of all the evil deeds and bend all his energy towards concentration of Divine Manifestation, when the seeker attains this noble perfection, then Allah Himself becomes the Guardian and Protector of the heart of His servant and provides light and guidance from the Divine treasures of enlightenment.” [al-Ghazali, Ihya Ulum id Din, vol. 3]