Facebook question, what does friends with mean?

[Problematic Problematic]

recently a friend of mine temporarily removed me as a friend from Facebook.

When I looked at my notifications, all the ones that came from him during that time had changed.

Instead of a typical notification I saw that it had this other Facebook friend of mine, who is a girl, listed next to him.

she isn't the only mutual friend I have in common with this guy, but all of the notifications listed her.

when he and I became friends again, the notifications returned to normal.

what does the friends with thing mean?
already posted the question their but the forums haven't given me a response just yet.