Let's see if I can muddy this up some.
The term 'gravity' specifically refers to the classical sense of it, the way Newton described it. He observed that two masses attracted each other and concluded a force must be acting on them. Newton did not know the cause of this force nor did he attempt to limit its speed, as this caused his equations to break down. This presents two problems: This applies only to objects with mass and the universal speed limit, the speed of light, is violated.
What we see on the earth is that the apple falls from the tree and accelerates to the ground. We assume we are not moving, since it sure looks that way. Therefore, there must be some force acting on the mass to cause this acceleration (F=ma).
What Einstein realized is that we are not at rest. Space is curved, as in the trampoline example from before. All objects move in space, and thus we follow the curvature of it. The apple is following this curvature to the center of the Earth. When it reaches the ground, it can't go any further since the Earth is in the way. The apple is constantly accelerating into the ground, but the ground pushes back on the apple. This is what gives us weight.
No force is needed in order for the apple to accelerate towards the ground, since F=ma does not apply in our accelerating frame of reference (we are constantly accelerating into the ground and the ground pushes back on us). Gravity can be thought of like this:
Imagine you are driving in your car, you take a ball and toss it up towards the roof. You quickly turn the steering wheel and watch the ball fly out towards the passenger door. From where you are sitting, there is not much movement going on. You are stationary in your car but the ball just shot towards the door, so there must have been some force acting on it. However, if you looked at it from a point of reference that was not accelerating (as in outside the car), you would see the ball going in a straight line and it was the car that moved towards the ball, thus, no force was necessary.
This is what happened with gravity. We are in this accelerating reference frame that we claim to be still. This gives rise to imaginary forces (gravity, Coriolis, centrifugal), just like the one the driver sees acting on the ball inside the car. If you remove yourself to a frame that is still, you would see there is no force at all.
Einstein's General Theory of Relativity completely changed what we thought happened to the apple. Gone were the forces and in their place was simple geometry of space. These deformations of space would carry at the speed of light and would be caused, not only by mass, but also energy and momentum. It would affect every object that traveled through this spacetime.
There is evidence GR is correct, it predicts the gravitational time dilation, in that time slows in higher gravitational fields, light (which has no mass) will be affected by objects with mass in space, and, because the limit on the gravitation is the speed of light, it correctly calculates the precession of the perihelion of Mercury.
So in the end, gravity is not a force, it only appears to us as one, because we are not looking at ourselves in the correct reference frame.
I sure hope that made sense, it's late.