It starts with a simple idea: there is no difference between a force due to acceleration and a force due to gravity. Albert Einstein showed that this is the case if spacetime (he had already shown that space and time were really a common entity in his theory of special relativity) is curved, and massive objects curve spacetime in their vicinity.

Pretty soon, you have black holes, wormholes, and all manner of weirdness.

Slendro is almost right about the acceleration/gravity thing, but he's a step or two too far foward in the theory. One of the implications of general relativity is that gravity is not a force, and acceleration doesn't exert a force anyway.

The theory of general relativity is based on the theory of special relativity along with Minkowski's geometrical formulation of it. It can be summed up in the equation E=mc², where E is energy, m is mass, and c represents the speed of light.

As I explained in the theory of relativity (go there to check the background info on any assertions I make here), the theory (the special one) is based on two principles:

1. "The laws of physics should work across all inertial frames of reference" and
2. One law of physics is that the speed of light, c, is a constant

Now, from inside any given frame of reference, it is assumed to be impossible to tell what the state of motion of that frame is. In a frame accelerating at a constant rate, anyone in that frame will be 'pushed' in the opposite direction to that of the acceleration by a 'force'. Try this in an elevator - when it starts off, note what you feel. When it accelerates up, you feel heavier, when it accelerates down you feel lighter. Incidentally, if you find a good elevator, jumping in the air while it accelerates downwards is a lot of fun, I've got up to 4 seconds air time this way, then got destroyed as it decelerated when I landed.
This 'force' is indistinguishable from gravity. I call it a 'force', but it isn't really. It's just a manifestation of inertiality. An explanation follows.

Inertia is the principle, observed throughout human history but not systematized until Newton, that objects do not change their state of motion unless an outside force acts on them. Objects travelling in inertial motion (motion with no forces acting on them) should continue to move in straight lines. F=ma. Because of this, and because objects were observed to be accelerated towards massive objects (the centre of the Earth, for one. Or the Sun, or whatever), Newton thought of gravity as a force.

However, this is because Newton thought of the universe as 3-dimensional. One of the consequences of Minkowski's geometry (see here) is that we must not think of the laws of physics as applying to a 3-dimensional 'space', but rather of a 4-dimensional spacetime structure. Newton only thought of gravity as a force because he perceived acceleration of objects in three dimensions as being non-inertial motion. However, in 4 dimensions, accelerated motion may be regarded as being inertial. That is, in 3-d an object can change its spatial position over time in inertial motion, add another dimension and an object can change its rate of change of position over time and still be in inertial motion. Phew, that was a conceptual challenge...

Ok, so objects in spacetime can accelerate and still be in inertial motion. This means that instead of having to view gravity as a force, it can instead be viewed as a property of spacetime... this is getting too abstract, no? Ok, let me put it this way. I'll stop trying to explain and just say that in order for spacetime to have this property, it must be curved. That's curved in 4 dimensions, not 3. Look, I don't know, take some psychedelics and get back to me, ok? In Newtonian Physics, objects in inertial motion travel in straight lines at constant velocities. In spacetime, objects can't travel in straight lines because the 'stuff' that the universe is made of is curved, so they have to follow curved paths. Just as a line is the shortest path between two points in a flat medium, the shortest path between two points on a curved surface is called a geodesic, and these are the paths that objects in inertial motion follow under general relativity. What we see as acceleration is actually an object following the shortest path in a set of dimensions we can only see 3/4 of.

How does this relate to gravity? Well, I've said that spacetime is curved, right? It's curved by the presence of mass, kinda like a rubber sheet is curved when you put something heavy on it (only in that case, gravity explains the curvature. In this case, the curvature explains gravity). The geodesics in the vicinity of Earth, for example, all curve in towards the centre of the planet, just like the sheet curves in towards that steel ball I told you to put on it.

The force you feel on your feet (well, probably more like on your ass, cos you're sitting down reading this) is not holding you 'stationary' by countering the 'force' of gravity, rather it is actually accelerating you off the inertial course you'd otherwise follow to the centre of the earth. It's pushing you around in spacetime.

Anyway, onto black holes. Spacetime can be curved by the presence of matter, yes? Loosely speaking, the more mass in an area the greater the curvature of spacetime in that area, ok? If you put a whole lot of mass all in one place, it curves spacetime and so it all falls in towards the other mass, and spacetime gets more radically curved and so it falls faster and closer and spacetime curves more until... infinite curvature. The graph you could draw of the curvature has a spike here, a spike that goes up forever. There is no spacetime as such in this region, the laws of physics no longer work in the singularity that has been created (general relativity is cool in that it predicts an exact circumstance in which it no longer works. In the singularity, the universe as we know it does not exist). For more on black holes visit the appropriate node.

Actually, the fundamental equation of General Relativity is NOT E = mc^2 as some people seem to believe, but rather Einstein's equation, G-alpha-beta = 8*Pi*T-alpha-beta, where G-alpha-beta is the Einstein tensor and T-alpha-beta is the Energy-Momentum tensor. The spacetime metric follows from G-alpha-beta, thereby defining the geometry of spacetime.

I see a big discussion breaking out here, and although I don't like jumping on the bandwagon for these things, I'll try to say a few things to clear up the confusion that aozilla has stepped into. First, a mildly technical explanation of the theory:

So you know that spacetime is a Riemannian manifold (ie. a space idetitified by a set of coordinates, usually space and time), and that the metric tensor allows you to measure distances within it. Now, where the real Physics comes in is when you have derivatives of the metric:

• First order effects come from first derivatives of the metric. Acceleration and Newtonian gravity belong in this category: in Special Relativity, acceleration is simply d²x^a/d²τ - the natural extension of what acceleration is in the Newtonian sense. In General Relativity, some extra terms quadratic in the velocity come in, and acceleration is then given by
  d²x^a/d²τ + Γ^a_bcdx^b/dτdx^c/dτ
  The equation come from specifying that freely falling (ie. unaccelerated) particles pursue geodesics. Don't worry too much about what the Γ term means, except to bear in mind that it is a thing expressed in terms of the first derivative of the metric. Now, because of all the factors of c knocking around, at low speeds we can forget about most of these terms that have just been added, and what we're left with as an expression for the acceleration the usual d²x^a/d²τ minus the acceleration due to the (Newtonian) gravitational field - and thus Newtonial gravity is recovered at small speeds in weak fields.
  It is which recovers the equivalence principle: that gravitational fields behave like accelerations.
• Second order effects come from the second derivatives of the metric, which naturally characterise the curvature of spacetime (mathematically speaking, these effects come from the presence of the Riemann tensor). Thus, the Einstein field equations - which involve the presence of matter - are expressed in terms of second derivatives of the metric (which quantifies the statement 'matter bends spacetime').

It is a fundamental principle of GR that we can always find local coordinates such that the first order effects vanish - that is, a frame can be accelerated such that the effects of gravity can be anulled.

Now to resolve aozilla's problem: he states that because of the inverse square law, gravity can be distingished from uniform acceleration. The key to the problem is the crucial requirement above that (in general) you can only makes the gravity balance the acceleration locally - as soon as you start thinking about inverse square laws, you are examining the particle's behaviour in a non-local context.
Indeed, if we could make accelerations equivalent to gravitational effect everywhere, then the first derivatives of the metric would vanish everywhere, and then so to would the second order effects. If this were true, then the Einstein field equations would fall apart and there would be no gravity anywhere.

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aozilla replies:Does this imply that the gravity is in essence coming from an infinitely far away source? The gravity is not within the system, but we still can measure exactly how far away the center of mass is...

What I was trying to say above is that the source is not important - the presence of matter produces seconds order effects (curvature) which are not relevent in as far as the Equivalence principle is concerned.
The fact that the gravitational field varies with position (whereas an uniform acceleration does not) is something that you must be observing on a 'larger scale' than that for which the Equivalence principle applies, so how is varies not important inasmuch as your question is concerned.