Relationship is a score that shows you close you are to people, either Students or Instructors.
It has a range from -10 to +10, with +10 being very good friends, while -10 will be an archenemy.
A good relationship will make it more likely that a student is willing to join your Clique and it will also determine whether you can use their specific adventure.
A good relationship with Instructors can be used for getting favors or extra lessons.
A bad relationship is needed in order to be able to duel with a student by declaring a Vendetta.