Moons (also called satellites) orbit around planets because the force of gravity keeps them in place. Once an object is held in place by gravity, it can not escape unless an exterior force acts upon it. Just as moons are held in place by gravity, planets are in turn held in orbit around stars by gravity, and stars orbit the galaxy. The amount of time it takes a moon to go around the planet it is orbiting is called a "revolution."
Isaac Newton described the relationship between two objects and what causes one to be caught in the gravitational pull of another. The equation (known as the Law of Universal Gravitation) is "F = GM1M2/d2," where "G" is the gravitational force constant (6.7x10-11 Newtons meter2 / kilogram2), the "M"s are the respective masses of the two bodies involves and "d" is the distance between the center of the bodies. Thus, the more massive one of the objects is, and the closer they are to one another, the large the force of gravity.
Published simultaneously in 1687 with the Law of Universal Gravitation, Newton's Three Laws of Motions also help explain the relationship between planets and moons, particularly the first and third law. The first law states that an object in motion tends to stay in motion until something acts on it, explaining why moons stay in their orbits. The third law states that for every action there is an opposite and equal reaction; thus, a moon exerts as much gravity on the planet as the planet does on the moon, and this is the cause of tides on Earth.
Newton's theories on gravity were accurate in the sense that they came up with correct results; however, his reasons for what caused gravity were later contradicted by Albert Einstein's General Theory of Relativity. Einstein showed that the cause of gravity is not necessarily an object's mass, but instead it is caused by the curves it makes in space-time. While this is difficult to visualize, his theory was later proved correct when it was shown that an object's mass can affect light waves, which would be impossible under Newton's theories.