The force that keeps on pulling all things down is called gravitational force. That is why no matter how hard you throw a stone up, it will fall back to the ground in a matter of seconds. This particular force originated from our planet Earth. Every planets, moon, stars and rocks on space has some degree of gravitational force. The more massive an object the stronger the gravitational force it will possess. This Law of Physics is also known as "Newton's law of universal gravitation".
This is the same force that keeps the moon on its orbit around the earth. In fact, the moon is falling on earth at the same time its escaping due to centrifugal force due to its revolution on earth.
As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.
As an analogy, Let us imagine you have bucket of water tied to a string. And you swing the bucket of water around around you.
As a bucket of water is tied to a string and spun in a circle, the tension force acting upon the bucket provides the centripetal force required for circular motion.
The string represents the gravity that balance (holds) the bucket from flying away because of centrifugal force you have applied to it (swinging).
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