When Sir Isaac Newton formulated his law of gravity in the 17th century, he called it the law of universal gravitation because he believed it applied to everything, everywhere. According to this model, gravity is thus a force that exists between any and every two objects in the universe. For example, your right shoe and the star Aldeberan --- about a million billion miles away --- exert a gravitational attraction on each other, however slight.
Since your right shoe doesn't fly into the sky and race towards Aldeberan, the force between your shoe and the star must be pretty small. Newton described the mathematical form of this gravitational force as being proportional to the masses of the two bodies and inversely proportional to the square of the distance between the two bodies. The equation that describes this is written as F = G x M1 x M2 divided by the distance squared; where F is the force, G is a number called the gravitational constant, and M1 and M2 the masses of the two bodies. So, for example, the planet Jupiter is pulling you toward it, but the Earth is pulling you toward itself about 40 million times more strongly.
Newton invented calculus to calculate the force of gravity. Since every object attracts every other object, then part of your backyard is attracting part of the Mare Crisium on the moon. Newton's calculus provides a way of adding up all the little pieces of the Earth and all the little pieces of the moon and figuring out how they all act together. What he figured out was that the Earth and moon influence each other just the same as if all of their mass was concentrated in one tiny point at each of their centers.
If you throw a rock as hard as you can, you can get it to go quite some distance before it falls back to the Earth. But the Earth's surface is curved. What if you could throw the rock so hard that in one second it had gone so far that the curve of the Earth made the surface "fall away" exactly that same distance? The rock could keep falling each second, but as long as it speeds fast enough to make the Earth's surface "fall away" that same amount, the rock would stay up forever, circling the Earth. That's what Newton realized was happening to the moon, and that's what happens to all satellites of the Earth --- they are constantly being pulled by the force of gravity.
Newton used the exact same equation to explain everything from orbiting planets to falling apples. Everything you will ever come in personal contact with operates according to Newton's law of universal gravitation. But when an object gets really heavy, or travels extremely fast, Newton's law doesn't work that well. Einstein figured out that Newton's law appears to be true because every object bends the space in which it sits. This formed the basis for quantum physics and other modern theories. Things get really complicated from there --- really, all you need to know for now is that Newton's law works almost all the time.