What is the relationship of the moon to the earth?

The moon and Earth are each affected by each other's gravity and rotate around a center of mass which is actually within Earth, since Earth is far larger in mass.

Because of the Earth's large force of gravity on the moon, the moon has undergone synchronous orbit i.e. always faces the same face to the Earth.

On the other hand, the moon influences the Earth by causing tides on Earth (two high and two low tides a day as well as larger pattern spring/neap tides).