Centripetal force:
The force which makes an object  in a circular path is known as Centripetal force.
Earths gravity on the moon, spinning a stone on a string.
Movement of the moon around the earth
The movement of the moon around the earth is due to the centripetal force. The earth provides the centripetal force. If there were no such force between the moon and the earth, the moon would continue a uniform, straight-line motion. It can be observed that falling objects are attracted towards the earth.
Does the apple attract the earth?
If so, we do not recognise the earth moving towards an apple. Why?

By the third law of motion, the apple also attracts the earth. But according to the second law of motion, acceleration is inversely proportional to an object's mass for a given force. The mass of an apple is negligibly small when compared to that of the mass of earth. So, we do not perceive the earth moving towards the apple.
Analyse why the earth does not move towards the moon?

In our solar system, all the planets revolve around the Sun. By arguing the same way, there must be a force between the Sun and the planets. From the above facts, Newton found that not only does the earth attract an apple and the moon, but all the objects in the universe also attract each other. This force of attraction between different objects is called the gravitational force.
Sun pulls all the planets and thereby provides a centripetal force on all the planets.