Why doesn't a global frame of reference exist for GR? I only have at best a layperson's familiarity with GR, so forgive me if I am asking a basic question, but I have heard that in GR, we cannot have a global frame of reference, that is a frame of reference that is applicable to the entire universe. Can anybody explain why this is? 
I have heard that it has something to do with the expansion of the universe, since we can still define it locally. At the same time I have also heard that it is due to our understanding of space-time as being curved. Can anybody expand on that?
 A: You use the term frame of reference but we need to be careful what we mean by this. In special relativity this phrase generally means an inertial frame i.e. a frame in which Newton's first law applies. In GR we obviously can't have a global inertial frame because objects accelerate (due to gravity) whenever they are near a mass so their behaviour isn't inertial and doesn't obey Newton's first law.
The more appropriate meaning of frame of reference is as a coordinate system i.e. any system of coordinates that we can use to describe positions in spacetime. In fact we can almost have a global coordinate system but it will almost always contain singularities that break it.
To make a simple analogy, consider the coordinate system we use to describe positions on the Earth i.e. the longitude and latitude. Now ask what is the longitude of the North and South poles? The problem is that all the lines on longitude meet at the poles, so the longitude is not defined there. Our coordinates are singular at the poles. There's nothing singular about the Earth at the poles, the problem is with our coordinates, but it means our coordinates cannot cover the whole Earth because they don't work at the poles. A similar (if rather more complicated) thing happens at the event horizon of a black hole.
Coordinate singularities can be removed by changing the coordinate system, however in general relativity we also encounter singularities that are not due to quirks of the coordinates. For example the singularity at the centre of a black holes is a true singularity and no system of coordinates will work there. So all the coordinate system we can devise wll break at the singularity.
You mention the expansion of the universe - this has a singular point at the Big Bang i.e. at $t = 0$. At this point we get the rather surprising result that the spacing between every point in the universe is zero. Again, no coordinate system will work at this point.
Strictly speaking we exclude singular points from the manifold that we use to represent spacetime i.e. the universe consists of everywhere except the (true) singularities. With this definition we can have a global coordinate system because the places where it breaks down are excluded by definition.
A: In general relativity there might not be a general frame of reference that will look the way an inertial frame of reference looks in special relativity. And the fundamental deep down reason is that we didn't assume there had to be, thus it didn't have to happen. Whether a particular solution to Einstein's equation has one or not is up to experiment to determine.
Why don't they exist or what is happening differently to make us say they are not there? Specifically you can't just take partial derivatives of things with respect to your coordinates the way you do in special relativity. Because coordinates in general relativity are only part of the story you actually use the metric in a nontrivial way to get your results.
And while in special relativity any frame moving at uniform velocity relative to an inertial frame was also inertial and so just as good this easy way to get just as good frames does not hold up as simply in general relativity.
General relativity gives you two options. Option one is to work with any coordinate system whatsoever and use the metric tensor in that coordinate system before you get your results. Or else option two is you can work with frames defined over a small region (of space and time) parameterized by a factor h where the metric tensor is very very close to the special relativity metric so you can compute your results like in SR up to errors second order in h (so can make them as small ad you like by making h small enough). And if you do the second option then indeed you can pick many frames in each small region (for each parameter h) that differ from each other by the standard SR Lorentz transformations (up to an order in h).
Now lets get to the things you've been told. Expansion has nothing to do with it, absolutely nothing. The curvature isn't actually really the issue either. If the universe was like a pac-man universe (go around in the x direction and come back to where you are but everything locally looks like ordinary flat space) you might fail to have a global coordinate system even though your space can be flat. And yes the equations of GR allow the universe to be flat everywhere and yet if you travel out in a region you come back to where you were. When you say a manifold is flat that doesn't mean it wouldn't look curved if you embedded it inside a larger dimensional thing (that is extrinsic curvature which depends on your embedding), it is about intrinsic curvature which is totally different. For instance the surface of a sphere is positively curved and you can tell that because if you try to sew a patch onto it you need to cut parts away. And the surface of a saddle is negatively curved and you can tell this because if you try to make a covering for a saddle with flat fabric you'd need to cut slits into the fabric and add more fabric to the slits. But the surface of a cylinder is actually truly flat (intrinsically). That's just a scalar measure of curvature (positive, negative, or zero scalar curvature) and the full measure of curvature comes from how vectors transported (virtually) along loops change. And there are 4*3/2=6 independent loops you have to consider in a 4d spacetime (n*(n-1)/2 in general) and you have to specify this for each of the four (n)  independent vectors and since they can change into a new vector you have to specify how the virtually transported vectors compare to each of the four (n) original independent vectors so you need 4*4*4*3/2 (nnn*(n-1)/2) numbers to fully give the geometry of curvature (actually there are some other symmetries so you don't need quite that many, nn(n*n-1)/12  so only 20 numbers for our universe). But the real point is that even if every vector around every loop can make the same angle with the loop and end up back to itself (a local property) it is still possible to end up back where you started if you make a long trip.
That said, you can use a single coordinate system for your whole spacetime in many situations people might want to say you can't. For instance you can have a spacetime with no matter and no fields other than gravity with a gravitational wave going in say the x direction. It is curved but there is a global coordinate system. You could even put an electromagnetic field going in the same direction and so have a spacetime that isn't empty. And these are actual solutions that have been found not just hypothetical solutions.
You can have an expanding universe with a coordinate system of (x,y,z,log t) with where (x,y,z) are the usual FLRW coordinates and t is the usual FLRW time with t=0 the big bang. This coordinate system covers everything that can be covered. If you used the coordinate system (x,y,z,log (t-1)) then you'd miss all the things that happened before one second. But with the (x,y,z,log t) coordinate system there is a curvature invariant that blows up as log t goes to negative infinity so we can't extend our coordinate system any farther.
This is what true singularities are. They are curves of finite metric length (or finite metric time) that run all the way out of your coordinate system in that finite length (or finite time) and have local invariants of things that can be measured blow up as you go out of the coordinate system so we know your coordinate system could not be extended.
If it took an infinite amount of time or space to leave your coordinate system that would not be weird, and if it took a finite length you'd think you just did it badly and left something out but if a local invariant blows up on the way out then it was unavoidable. And an invariant is a measurement that is finite when the curvature is finite but doesn't depend on the coordinate system you use.
So you just have to use the metric tensor. And curvature isn't a problem (you now have examples with curvature with global coordinates) and expanding space isn't a problem (you have an example with a single coordinate system that can't be extended). And sometime there isn't a global coordinate system (for pac-man style reasons) that can happen even when all 20 independent curvature components are zero everywhere, no curvature.
The real reason is that we made a theory about local measurements. So we gave ourselves the freedom to have systems without a global coordinate system. Whether we needed that freedom depends on what we see in the universe, if we see evidence of pac-man we can handle it without changing the theory.
In summary @MikeH I'm saying that for local measurements there are frames that approximately just like the frames of SR. And that globally there are frames that are different, for instance inertially moving bodies that start out tangent could move closer towards each other. If you accept these frames as normal then the only known reason you can't use them is if the universe has weird things like wormholes or groundhog days or a pac-man style space. For all of those you could say that you just end up in a universe that looks the same but is actually different, so I'm not sure you'd ever be forced to use local frames.
A: Actually, a global coordinate system is possible. Most treatments of GR rely on the concept of a manifold, emphasizing that coordinate systems are (in general) only local. However, the Whitney embedding theorems show that any manifold can be embedded in $R^n$ for some positive integer $n$. Perhaps the original poster might not be offended if I tack on a strongly related question: Suppose we establish a local coordinate system and, from its origin, observe a particle moving through space. When the particle moves close to the boundaries of the local patch containing our origin, how do we continue to describe the particle's trajectory relative as it moves into the adjacent patch and out of our own? Of course this would not be a problem in $R^n$, but only if we insisted upon measurements only intrinsic to the original manifold.
