What is gravity and how do you define it? 

There are two levels at which to work. First, there is Newtonian gravity—an attraction that exists between any two objects and that pulls each object toward the center of mass of the other object with a force that's equal to the gravitational constant times the product of the two masses, divided by the square of the distance separating the two objects. For example, you are attracted toward the earth's center of mass with a force equal to the gravitational constant times the product of the earth's mass and your mass, divided by the square of the distance between the earth's center of mass and your own center of mass. This force is usually called "your weight." The earth is attracted toward your center of mass with exactly the same amount of force.
Second, there is the gravity of Einstein's general relativity—a distortion of space/time that's caused by the local presence of mass/energy. Space is curved around objects in such a way that two freely moving objects tend to accelerate toward one another. As long as those objects aren't too large or too dense, this new description of gravity is equivalent to the Newtonian version—they both predict exactly the same effects. But when one or both of the objects is extremely massive or very dense, general relativity provides a more accurate prediction of what will happen. In reality, mass/energy really does warp space/time and general relativity does provide the correct view of gravity in our universe. The next level of theory, quantum gravity (which will reconcile the theory of general relativity with the theory of quantum physics), is still in the works.


