Does the curvature of spacetime theory assume gravity? Whenever I read about the curvature of spacetime as an explanation for gravity, I see pictures of a sheet (spacetime) with various masses indenting the sheet to form "gravity wells." Objects which are gravitationally attracted are said to roll down the curved sheet of spacetime into the gravity well. This is troubling to me, because, in order for objects on the locally slanted spacetime sheet to accelerate, gravity must be assumed. Therefore I ask; does the explanation of gravity as the curvature of spacetime assume gravity? If yes, what is the point of the theory? If No, what am I missing?
 A: Those sheets with dips having stars at their centers illustrate gravity by illustrating that freely falling objects move along geodesics, and that geodesics are curved by the dips in a way that looks as if freely falling objects were attracted by the stars. (Needless to say, the 2-dimensional sheets remain poor substitutes for the real McCoy, which is a (3+1)-dimensional pseudo-Riemannian manifold.)
A: The rubber sheet model is not meant as an explanation (proposed underlying mechanism) for anything. It is only a way of visualizing the gravitational interaction of objects in space in terms of something more familiar. We are all familiar with the motion of objects on slopes in a uniform gravitational field, so it's a useful toy model.
It is not a model of the curved spacetime of general relativity. See "General relativity versus the rubber sheet" below.
Newtonian gravity can be described in terms of a potential field, which has a value (a real number) at every point in space. Objects experience an acceleration that is equal to minus the gradient of the field. That is, they accelerate in the direction that the field around them is decreasing most rapidly, and the magnitude of the acceleration is proportional to the slope (derivative) of the field in that direction. Also, the field satisfies Poisson's equation, which allows you to find its value at every point if you know the distribution of matter.
Objects on a hill on Earth likewise roll downhill, and the formula for the acceleration in terms of the local elevation is the same (subject to idealizing assumptions) as the formula for the gravitational acceleration in terms of the local potential. Not only that, but the deformation of a horizontal elastic sheet in a uniform gravitational field with heavy objects resting on it is described (approximately) by Poisson's equation. The lower the mass density of the sheet, the vertical stretching, and the friction, the better this toy model approximates Newtonian gravity.
One big caveat is that a 2D rubber sheet is described by the 2D Poisson equation, which has different solutions than the 3D Poisson equation. With a 2D sheet you get 2D Newtonian gravity, which has a 1/r force law instead of 1/r². So in this sense it is far from correct. This problem only affects the elastic sheet part, not the rolling-downhill part. If you construct a rigid surface with the correct shape from the 3D Poisson equation, like the "gravity wells" found in some science museums, you get a pretty accurate simulation of test particles in a 1/r² gravitational field.
On to general relativity.
General relativity versus the rubber sheet
In general relativity, the motion of objects depends on the intrinsic shape of spacetime.
You can embed hunks of curved space into Euclidean space. Like the rubber sheet analogy this is just a way of understanding them in terms of something we're more familiar with (in this case, space that isn't curved).
These embeddings do not have any "down" direction. An ant crawling along the surface (which is a much better analogy to the effect of curved spacetime on test particles) doesn't care about ambient gravitational fields. Ideal ants (much like real ants) are as happy to crawl on the ceiling as the floor. They will follow the same path however the surface is oriented, in stark contrast to the rubber sheet (or a rigid gravity well) where turning the surface upside down turns an attractive force into a repulsive force.
It would make sense to always show GR embeddings as "hills" instead of "valleys", since it makes no difference to the physics and it would avoid unnecessary confusion with gravity wells. For some reason, not only is this not the standard, it is almost unheard of in works for popular audiences. Either the authors want to invite confusion, or they don't understand the difference themselves. I suspect the latter.
A: A better analogy for curvature is to imagine ants walking on bowling ball in space.  They start out at the same point exchange information in some ant fashion, then take off in different directions and even if they try to walk as straight as possible on the surface, they end up converging towards that point opposite where they started (And if you were at the north pole and started flying as straight as possible at constant altitude, you head to the south pole, no matter which direction you headed out).
Those kinds of things are what the curvature is supposed to do, to create paths that are the natural ones that converge, and that converge in a way that didn't depend on say if you were small and light or if you were more massive, just the path you take.
That said there is an important further point, which is that curvature exists even far from sources because curvature begets curvature. For instance, mass, energy, momentum, stress, and pressure are sources of curvature, but they are not the only things that create curvature, curvature itself can create further and additional curvature. A gravitational wave can propagate or even spread in a vacuum of empty space devoid of all mass, energy, momentum, stress, and pressure.
The region outside a symmetric nonrotating static star is curved, even the parts far from any mass or energy or momentum or stress or pressure. The space remains curved because the existing curvature is exactly shaped so as to persist (or otherwise cause future curvature exactly like itself).
So curvature allows and sometimes requires more and/or future curvature, just as a travelling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would.
Imagine a flat region of space shaped like a ball, then imagine a funnel type curved space where two regions of surface area are farther apart than they would be if flat (like a higher dimensional version of a funnel and on a funnel surface two circles of a particular circumference are farther away as measured along the funnel then if two similarly sized circles were in a flat sheet). On its own, spacetime doesn't allow itself to connect those two kinds of regions together, but that mismatch is exactly the kind or not-lining-up that putting some mass or energy right there on the boundary fixes. So without mass those two regions can't line up, with mass they can. Just like an electromagnetic field can have a kink if there is a charge there.
So your curvature likes to propagate a certain way, and if you want it to deviate from that, you need mass, energy, momentum, stress, and/or pressure. And you'd need the right kind to get it to match up, the kind you want might be available, and might not even exist, so not all kinds of curvature will be allowed. But the point of a source is that it changes the balance between nearby curvature and not that affects future curvature. So there is a kind of balance, and there are things that can warp that a balance. Those things that warp that natural vacuum balance are called gravitational sources.
So that means you want to depict two things, firstly nearby curvature affects nearby curvature, that current curvature determines future curvature, and secondly that gravitational sources allow the curvature to be different than what it would be on its own.  The curvature itself is supposed to remind you about paths converging regardless of whether a light thing or a heavy thing sets out on the path.  But when you look at the analogy for gravity look for those three features, but don't take any other aspect of the analogy too seriously, the curvature of time is important and not pictured, and the curvature of space is different than as pictured, and things don't move within the space because of an external space or because of an external force and finally the sources themselves make different curvature by allowing pieces that otherwise wouldn't fit to fit together. 
A: I greatly sympathize with your question. It is indeed a very misleading analogy given in popular accounts. I assure you that curvature or in general, general relativity (GR) describe gravity, they don't assume it. As you appear to be uninitiated I shall try to give you some basic hints about how gravity is described by GR.
In the absence of matter/energy the spacetime  (space and time according to the relativity theories are so intimately related with each other it makes more sense to combine them in a 4 dimensional object called space-time) is flat like a table top. This resembles closely with (not completely) Euclidean geometry of plane surfaces. We call this spacetime, Minkowski space. In this space the shortest distance between any two points are straight lines.
However as soon as there is some matter/energy the geometry of the surrounding spacetime is affected. It no longer remains Minkowski space, it becomes a (pseudo) Riemannian manifold. By this I mean the geometry is no longer like geometries of a plane surface but rather like geometries of a curved surface. In this curved spacetime the shortest distance between any two points are not straight lines in general, rather they are curved lines. It is not very hard to understand. Our Earth is a curved surface and the shortest distance between any two points are great circles rather than straight lines. Similarly the shortest distance between any two points in the 4 dimensional spacetime are curved lines. An object like sun makes the geometry of spacetime curved in such a way that the shortest distance between any two points are curved. This is called a geodesic. A particle follows this curved geometry by moving along this geodesic. Einstein's equations are mathematical descriptions of the relation of the geometry to the matter/energy.
This is how gravity is described in general relativity.
A: No. While the curvature of spacetime -- or even Newtonian gravity, for that matter -- indeed can be modeled as a "potential well", the tendency of matter to lower this potential is an axiom of general relativity, and is not gravity. 
The mathematics of general relativity can be derived from four important physical axioms -- (1) the Einstein-Hilbert action, or "gravity is the curvature of spacetime", or equivalently the Einstein-Field Equation, "matter curves spacetime" -- see my answer here for a derivation of the EFE from the action, (2) the geodesic equation, or "the geometry of spacetime moves matter", (3) Newtonian gravity is effective at low energies and (4) special relativity. So while it is true that general relativity assumes some law on whose basis matter moves (the geodesic equation), this law is not "gravity".
A: I will admit that my understanding of General Relativity at the moment is limited, but this is my understanding:
When dealing with curved coordinate systems, like space-time curved by a massive object, you can calculate what is known as a "geodesic" which is a straight line in curved coordinates. If you are dealing with a spherical space, all geodesics will be "great circles" which wrap around the circumference of the sphere. While this will be the shortest path between 2 points, it can seem oddly curved to anyone on the sphere (if you're confused about this than here is a program someone made that should help explain.)
Where this comes into general relativity is that objects will follow geodesic paths through space-time, and as you may have learned: everything must move through space-time at the speed of light which is why when you move at the speed of light then you experience no time because all your motion is through space and none of your motion can be through time. A massive object will create what is basically a shortcut through time, so objects will move through space as it is a shorter path.
With this in mind, feel free to refer to any form of descending whether it be (walking down stairs, jumping down or taking an elevator) as "taking a shortcut to the future".
A: Gravity is not as important to the rubber-sheet analogy as one might think. It is needed for two things:

*

*Causing the weights to curve the sheet

*Causing the weights stay on the sheet as they move

Anything that made these two conditions true would work for the demonstration.
In fact, gravity is not an ideal mechanism for #2 because it has a component in the direction of the sheet, and so it causes acceleration along the sheet that actual general relativity does not exhibit.
In any case, the rubber-sheet analogy only shows how space curves due to mass. In reality, most of the effects we observe are due to the curvature in the time direction. To visualize that, a cone- or horn-shaped curvature is more representative.
Here are a few visualizations for you:

*

*A 90-second illustration

*Vsauce "Which Way Is Down?"
Also Rickard Jonsson's thesis on how to visualize relativity:
http://www.relativitet.se/Webtheses/tes.pdf
A: Here's a picture of what you mean:

It might be obvious to make this happen, gravity has to pull on the objects to make them move on the sheet. So this visualization is truly misleading, to say the least! It doesn't show how the curvature of space (and just as importantly, time!) leads to geodesic motions of objects through 3-d spacetime. Maybe the objects follow a geodesic path on the rubber sheet, but that's the case in Newtonian mechanics too. That is, in Euclidean space (according to the principle of least action).
