This is partially a lesson in vocabulary, and partially a very deep fact about Einstein's Theory of General Relativity.
Gravitation is the word for the phenomenon of objects tending to fall into each other, orbit around each other, etc. This word does not imply a particular cause for the phenomenon.
The "force of gravity" is a term that's appropriate if gravitation is caused by a force. That is, the earth is somehow "pulling" on objects, and that's why they fall. This is how things work in Newtonian gravity, but not how they work in General Relativity.
Instead, in General Relativity, there is no force of gravity, and gravitation arises because spacetime (i.e. space plus time, considered mathematically as a unified entity) is curved. Things with no forces acting on them follow geodesics (the "shortest path," more or less) through curved spacetime. Objects cause spacetime to curve, which in turn affects the movement of other objects. Thus the earth is not "pulled in" by the sun; instead it travels along a geodesic that forms an ellipse because of the curvature caused by the sun in its vicinity.
Thus, according to legend and Newtonian physics, Newton made his discoveries when the earth's gravitational pull caused an apple to fall and hit him on the head. But according to Einstein, the discovery was made when an apple was released from its branch and, with no forces acting on it, followed a null geodesic through curved spacetime that would have taken it to the center of the Earth had Newton's head not gotten in the way.
Einstein's way of thinking, confusing though it may be, gives more accurate answers, although it hardly matters for apples under ordinary circumstances.
http://en.wikipedia.org/wiki/Gravitation
http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation
http://en.wikipedia.org/wiki/General_relativity
http://en.wikipedia.org/wiki/Spacetime
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Discussion (2)
This is very parallel to the way that centrifugal force is not a true force in Newtonian physics. Rather, it's an illusion caused by being in an accelerating (non-Newtonian) frame of reference.
In reality, whatever's making you rotate exerts a centripetal force of some sort, and it's your own inertia (i.e. natural tendency to keep going at constant speed in a straight line) that causes the centrifugal force.
Right. Gravity actually shows up as a "fictitious force" in General Relativity, arising from the fact that those standing on the surface of the earth are in a non-inertial reference frame. (Really! Because there's no gravitational force, and a geodesic would go to the center of the earth, anyone standing on earth's surface has only the normal force upward acting on them.) Like all fictitious forces, it is proportional to the mass of the object--which, from a certain perspective, can be viewed as a pre-Relativity clue that gravity isn't a true force.
I don't like talking about Truth, though--I prefer to talk about what's true in particular models and how accurate those models are.