The force responsible for the moon staying in orbit around the earth is a result of their gravitational fields interacting with each other. Earth being the more massive planet exerts a force on the moon pulling it towards itself, as a result of the third Newtonian law of motion, the moon exerts an equal and opposite force which enables it to remain in a stable orbit around the Earth. Another force resulting from this interaction is the centrifugal force which is likely responsible for the circular motion of moon's orbit around the Earth.
