I remember reading "The First Man in the Moon" by H.G. Wells quite a long time ago. Wells describes a machine with which a scientist travels to the moon. In this machine the scientist was pulled towards the rear end of the rocket (facing the earth) for the first two-thirds of the trip. At two-thirds he was momentarily weightless but after that he was pulled towards the front of the rocket (the side facing the moon). Now in this story, the rocket wasn't propelled by any real laws of physics, but it makes you wonder, what happened to Armstrong, Aldrin and Collins when they flew to the moon?
Well, clearly, when they took off and the engines were buzzing they were pulled hard towards the earth side of the rocket. But once in space, they turned off their engines. Was there a slight residual pull towards the rear of the rocket because the earth was much closer than the moon? Take a minute to think about this before you read on.
Gravitation pulls exactly as hard on the rocket as on the astronauts and therefore both their speeds decrease, but by exactly the same amount. So inside the rocket you won't notice a thing until you turn on your engines again (or hit an object in space.) The same thing happens to astronauts in the international space station. In this case, the space station is in an eternal fall towards earth. However, through its forward speed it keeps missing the earth! Since space station and astronauts fall with the same speed they are not noticing their fall.
But the story gets better. The gravitational force between two objects is F = -G M * m / R^2 where G is some constant, M the mass of the earth and m the mass of the space station or astronaut and R the distance between them. Thus, the force is stronger for the space station than for the astronaut because its mass (m) is much bigger. Now, when we compute the motion for the station or astronaut through space we need to equate the force with "m * a" where "m" is again the mass and "a" the acceleration. If this is hard to follow forget about the equations and remember this: the mass "m" drops from the equation and the orbit that one can now compute is independent of the mass m!
This magic cancellation is kind of coincidental or suspicious if you want. Perhaps there is a simpler principle at work here. Einstein thought so too. He imagined a version of the following thought experiment. Imagine you are knocked unconscious for a full year and wake up in an elevator. You feel a downward pull on your body and your company tells you this is because you have been abducted and now live on a bigger planet with more mass and thus more gravitational pull. He assures you the elevator is not moving. You are not so sure, because isn't a simpler explanation that you are accelerating upwards in a building? The point is that there is no way to tell the difference between gravitation and accelaretaion and Einstein concluded that if there is no measurement to tell the difference, well then there might be no difference... It's the Turing test for gravity!
In Einstein's general relativity there is no gravitational force and no mysterious cancellations. Mass and energy (remember E=M*c^2) bends space and time in such a way that what used to be straight lines become curved trajectories. Every object travels through this space in exactly the same way without any forces acting on it. The only thing that has changed is that a straight freeway became a roller coaster, but this change is identical for all objects traveling on it, big or small. You may only experience a "force" when you accelerate out of these "free fall geodesics". For instance, simply standing on the surface of the earth blocks the natural geodesic that would move to you the center of the earth. In this way, the earth accelerates you upward, as if you were firing the engines of a rocket. And so what you would classically call gravitational pull becomes upward acceleration, just as the man in the elevator.