Familiarity is the state of knowing something very well. It's also a casual way of behaving. Your familiarity with your best friend means that you can say whatever you want around her.
Familiarity is from the Latin word familiaritatem for "intimacy, friendship.” Familiarity is used in situations that people know well. A painter has a familiarity with brushes and canvas. When there’s familiarity with a task, people aren’t stressed, because they’ve done it before. This word also refers to a casual way of acting: if you hug someone, it shows familiarity. That kind of familiarity is the opposite of aloofness and formality.