General relativity says that energy (in the form of mass, light, and whatever other forms it comes in) tells spacetime how to bend, and the bending of spacetime tells that energy how to move. The concept of "gravity" is then that objects are falling along the bending of spacetime. The path that objects follow is called a "geodesic". Let's begin by looking at the bending side of things, and then we'll come back to look at geodesics.
The amount of bending that is induced by an object is directly related to that object's energy (typically, the most important part of its energy is its mass energy, but there can be exceptions). The Sun's mass is the biggest contribution to bending in our solar system. So much so, that it dwarfs the bending of spacetime by the Earth to the extent that to a very good approximation, we can just consider the Earth to be massless as it travels around the Sun (we call this the test particle limit). Similarly, when you're standing on the Earth, the Earth's mass dominates the bending of spacetime over your own, and so you can treat yourself as a massless test particle for all intents and purposes. However, truth be told, you warp the spacetime around you just a teensy tiny bit, and that does have an impact upon the earth in response.
Now, let's get back to those geodesics. A body undergoing geodesic motion feels no forces acting upon itself. It is just following what it feels to be a "downward slope through spacetime" (this is how the bending affects the motion of an object). The particular geodesic an object wants to follow is dependent upon its velocity, but perhaps surprisingly, not its mass (unless it is massless, in which case its velocity is exactly the speed of light). There are no forces acting upon that body; we say this body is in freefall. Gravity is not acting as a force. (Technically, if the body is larger than a point, it can have tidal forces acting upon it, which are forces that occur because of a differential in the gravitational effect between the two ends of the body, but we'll ignore those.)
OK, so let's look a little deeper into these geodesic things. What do they look like? Standing on the surface of the Earth, if we throw a ball into the air, it will trace out a parabola through space as it rises and then falls back down to Earth. This is the geodesic that it follows. It turns out that given the appropriate definition, this path is the equivalent of a straight line through four-dimensional spacetime, given the bending of spacetime. How does this relate to what we think of as the acceleration due to gravity?
Let us choose a coordinate system based on our location on the Earth. We'll say that I'm at the origin, and define that we throw the ball up in the air at time t = 0 (this is essentially giving a name to the location, nothing more). We can describe the position of the ball in spacetime in this coordinate system using an appropriate parameter (that we call an "affine parameter"). As the ball moves through spacetime, its position in spacetime is given by appropriate functions of this parameter. We can rewrite things slightly, to relate its position in space to its position in time. Then, when we look at this trajectory, it appears that the object is accelerating towards the earth, giving rise to the idea that gravity is acting as a force.
What is really happening, however, is that the object's motion in our coordinate system is described by the geodesic equation. If you want some maths, this equation looks like the following:
(geodesic equation) (image courtesy of http://en.wikipedia.org/wiki/Geodesic_equation#Affine_geodesics)
Here, x (with superscript Greek indices) describes the position of the ball in our coordinate system. The indices indicate whether we're talking about the x,y,z or time coordinate. The parameter t that the derivatives are being taken with respect to is the affine parameter; in this case, it is known as the "proper time" of the object (for slowly moving objects, we can think of t as the time coordinate in our coordinate system). The first term in this equation is the acceleration of the object in our coordinate system. The second term describes the effect of gravity. The thing that looks like part of a hangman's game is called a connection symbol. It encodes all of the effects of the bending of space time (as well as information about our choice of coordinate system). There are actually sixteen terms here: it's written in a convention called Einstein summation convention. This shows that the effects of the bending of spacetime change the acceleration of an object, based on its velocity through not only space but also through time.
If there is no curvature to spacetime, then the connection symbols are all zero, and we see that an object moves with zero acceleration (constant velocity) unless acted upon by an external force (which would replace the zero on the right-hand side of this equation). (Again, there are some technicalities: this is only true in a Cartesian coordinate system; in something like polar coordinates, the connection symbols may not be vanishing, but they're just describing the vagaries of the coordinate system in that case.)
If there is some bending to spacetime, then the connection symbols are not zero, and all of a sudden, there appears to be an acceleration. It is this curvature of spacetime that gives rise to what we interpret as gravitational acceleration. Note that there is no mass in this equation - it doesn't matter what the mass of the object is, they all follow the same geodesic (so long as it's not massless, in which case things are a little different).
So, what good is this geodesic description of the force of gravity? Can't we just think of gravity as a force and be done with it?
It turns out that there are two cases where this description of the effect of gravity gives vastly different results compared to the concept of gravity as a force. The first is for objects moving very very fast, close to the speed of light. Newtonian gravity doesn't correctly account for the effect of the energy of the object in this case. A particularly important example is for exactly massless particles, such as photons (light). One of the first experimental confirmations of general relativity was that light can be deflected by a mass, such as the sun. Another effect related to light is that as light travels up through the earth's gravitational field, it loses energy. This was actually predicted before general relativity, by considering conservation of energy with a radioactive particle in the earth's gravitational field. However, although the effect was discovered, it had no description in terms of Newtonian gravity.
The second case in which the effect of gravity vastly differs is in the realm of extremely strong gravitational fields, such as those around black holes. Here, the effect of gravity is so severe that not even light can escape from the gravitational pull of such an object. Again, this effect was calculated in Newtonian gravity by thinking about the escape velocity of a body, and contemplating what happens when it gets larger than the speed of light. Surprisingly, the answer you arrive at is exactly the same as in general relativity. However, as light is massless, you once again do not have a good description of this effect in terms of Newtonian gravity, which tells you that there has to be a more complete theory.
So, to summarize, general relativity says that matter bends spacetime, and the effect of that bending of spacetime is to create a generalized kind of force that acts on objects. However, it isn't a force as such that acts on the object, but rather just the object following its geodesic path through spacetime.
