Well, I don't have a deep level of understanding of it at this point (working on it, slowly, but it's not my primary focus at this time). But here goes with a pretty broad overview.
So, Einstein developed General Relativity in the 1910's, right? And it describes gravity in terms of the curvature of a spacetime manifold. That is, you have a 4-dimensional geometric space, with a (semi-Riemannian) metric tensor, which is to say, some way to measure the lengths of curves and the angles at which they intersect. (The "semi-Riemannian" part is because time is one of the dimensions, and it does not act the same as the three spatial dimensions. Curves in the "time" dimension end up having "negative length," whereas curves in the other dimensions have positive length -- or vice versa, depending roughly which coast of the US you're on, but let's not get down in the weeds). Associated to this metric (that is to say, determined by it, but not in a very obvious way) is a "curvature tensor," which tells you how spacetime is curving. And what we think of as gravitation is simply an effect of the fact that the metric is determined by the presence of matter in such a way that, when there's mass, the curvature nearby is not zero, so "straight lines" are not really straight, but rather curve toward the mass. (Kind of like how the earth is curved, so if you "drive in a straight line" on the earth, you don't drive in what we would think of as a straight line -- although like almost every analogy used to explain GR, this one is profoundly flawed). By the way, this is a little more complicated even than it sounds, because I just said that the curvature/metric was determined by where matter is, but of course, where matter is is determined by the curvature (gravity moves things), so it all gets quite complicated.
OK, so that's GR. In the 1920s, meanwhile, quantum mechanics gave a correct description of the behavior of atomic particles, such as electrons, and was generally spectacularly successful; but there were some things it couldn't handle very well, like describing how particles interacted with photons (which are also particles, after all), or generally how we should think of fields, like the electromagnetic field.
Now the thing is, there was an obvious thing to TRY, with the electromagnetic field, but whenever you tried to calculate anything in that way, you always got infinity for your answer, which wasn't very useful. People tried all kinds of increasingly crazy and exotic things to deal with this, but about 20 years later, some young physicists found that the answer wasn't too horribly exotic at all -- you just had to find a clever way to introduce another, negative infinity into your calculation, in such a way that the two infinities cancel and give you a finite number. If you think this sounds horrifying, then you're in the company of many mathematicians; but it's not actually so bad. The trick where you added another infinity (in a prescribed, systematic way so you get a well defined answer) to cancel the first one is called renormalization. The first major theory to come from this was Quantum Electrodynamics (QED), which finally gave a theory for the interaction of matter and light, and incidentally allowed incredibly precise computations of such things as the anomalous magnetic moment of the electron. (Actually, this is often cited as the most accurate prediction ever given by a scientific theory, as to date it's verified out to about ten digits). QED is an example of a quantum field theory -- a quantum theory where the objects of studies are fields. (The particles, like the photon and electron, emerge as "quanta" of the fields, or little local excitations. One of the great advances of QED/QFT over older quantum theory was that it gave the ability to study the creation and annihilation of particles, not just the behavior of particles that existed).
Now, a major dream since the 20s had been to "unify" electromagnetism and gravity, in the same way that electricity and magnetism had themselves been unified by Clark Maxwell: give a coherent theory where both emerged as different aspects of the same theory. By the 50s or 60s when QED was being wrapped up, though, nuclear physics had taught us about two new forces, the strong and weak nuclear forces. Over the course of the 60s and the 70s, both of these were successfully unified with electromagnetism in the framework of the Standard Model of Particle Physics. In each case, one of the big hurdles was finding a systematic way to deal with the infinities that crept up if you did the obvious thing -- although by this time, I am leaping over a LOT of detail, like gauge field theory, and a host of other things.
One thing worth mentioning is that these theories all developed as ways to calculate approximate answers from an exact theory that was too hard to get any answers from -- perturbative theories, they're called. In the case of QED, as we've seen, the approximation is spectacularly good. In the case of the strong force, it's far less good, and there are significant effects from the "real picture" that aren't included in the approximation, so that the answers people get are just close, not super exact. That doesn't mean there isn't an exact theory, just that it's extremely difficult to work with. (Though with computers, now, people are trying to do so numerically, and skip the analytical steps).
Anyway. Through all this time, and ever since, there has been one problem annoying physics more than any other: there is, as before, an obvious way to add the gravitational field (that metric, discussed earlier from Einstein's GR) into this picture. But it ALWAYS gives infinities, and there is simply no way on earth to renormalize it. Hence, there is no consistent theory of quantum gravity, which means among other things that, when studying the very dense, where scales are small but masses are large, both GR and QFT are relevant, and they give radically different pictures of reality, so we just don't understand black holes / the big bang / etc., that well.
OK, so finally, string theory. The reason, as I understand it, that people are really interested in string theory is that they discovered that, if you view elementary particles as one-dimensional strings instead of 0-dimensional points, so that your motions become integrals over a sheet instead of over a line, all the integrals that diverged (went to infinity) before remain finite now, and you can quite easily get gravity to play nicely with the other forces.
That's the good news. The bad news: for this all to work, you have to work in 10 or 11 dimensions, where 6-7 of them are "compactified," or so small we couldn't see them; unlike the other theories, in string theory we have *only* a perturtative/approximate theory: how it relates to any specific nonperturbative or "exact" theory is (so far as I understand) unclear; (relatedly) the mathematics are difficult beyond belief, and so very slow progress is being made on being able to predict *anything* from this theory. Making any predictions at all, in fact, is very hard, and any predictions that are made are at an energy scale many orders of magnitude (I think about 20-30) beyond anything that we can reach at our best particle accelerator (SSC). Moreover, there are 10^500 different equally consistent string theories (so far as anybody knows, anyway -- the math to say otherwise is beyond our reach, at least), and they give different answers/universes, so it's not even clear how one could test it even if the other issues weren't present. (Though there would certainly be features that should be present in all 10^500, I believe, but remain inaccessible for the conceivable future).
So I said before that in string theory, you could get gravity to play nicely with the other forces, and that's true; but the caveat is that it's quite difficult to tell whether those forces end up looking like the ones in our actual world at the end of it all. (Though I believe there has been at least one computation, black hole entropy, which did reproduce the classical result).
I'm not sure if that addresses any of what you wanted to know. My best guess would be no. Also, all of it should carry the usual disclaimer that I'm a mathematician, and a physicist may come along, read it, get very angry, and tell me what-for. I expect I would find the experience very educational.
Someone explain String Theory to me
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