The Inquisitorial Rank is the second form of progression in the game and it represents your standing within the Holy Order of the Inquisition. Unlike the Account Level, this is character based and tracks progression individually.
It can be increased by gaining Rank Points by completing Tarot missions and playing through the single player campaign.
The Inquisitorial Rank is the main factor that decides what parts of the Caligari sector are open to you and determines how powerful rewards you receive from the Ordo armoury at the end of missions.