Strict Definition: The force acting on a body from another body's gravitational field. ie: The feeling of your
weight while standing on the Earth's surface.
Equation: Starting with Newton's 2nd law, f=ma. (F: force, M: mass, A: acceleration). In the case of gravitation where the gravitational field of the affecting body is much greater than the affected body, A can be substituted with the acceleration due to gravity in that particular field. On Earth, this is 9.8 m/s^2 (meters per second squared) at the surface. (Yes, g changes based on your position on/above the Earth's surface!) Often, the gravitational acceleration is shortened with just g. Thus, for gravity, f=mg.
Common Definition: A term often used to express the magnitude of centripetal force associated in circular (or arc) motion, as a scalar multiple of the force of gravity. i.e.: 5G's would be a lateral force equal to 5 times the force of gravity.