I've noticed that none of these answers actually answer the question.
The simplest explanation of string theory I can think of:
Particles we currently consider "point particles" (electrons, quarks, photons, etc.) are actually tiny pieces of string with each a characteristic vibration. They interact in a sort of harmony that results in/manifests as the physical laws we observe.
If anyone with more knowledge in the field can correct me, I ask for improvements. This is just how I personally explain it to people who ask, and I'd hate to give out false information.
Here is the way I would try to explain Loop Quantum Gravity to my grand mother. Loop Quantum Gravity is a quantum theory. It has a Hilbert space, observables and transition amplitudes. All these are well defined. Like all quantum theories, it has a classical limit. The conjecture (not proven, but for which there are many elements of evidence), is that the classical limit is standard General Relativity. Therefore the "low energy effective action" is just that of General Relativity.
The main idea of the theory is to build the quantum theory, namely the Hilbert space, operators and transition amplitudes, without expanding the fields around a reference metric (Minkowski or else), but keeping the operator associated to the metric itself. The concrete steps to write the theory are just writing the Hilbert space, the operators and the expression for the transition amplitudes. This takes only a page of math.
The result of the theory are of three kind. First, the operators that describe geometry are well defined and their spectrum can be computed. As always in quantum theory, this can be used to predict the "quantization", namely the discreteness, of certain quantities. The calculation can be done, and area and volume are discrete. therefore the theory predicts a granular space. This is just a straightforward consequence of quantum theory and the kinematics of GR.
Second, it is easy to see that in the transition amplitudes there are never ultraviolet divergences, and this is pretty good.
Then there are more "concrete" results. Two main ones: the application to cosmology, that "predicts" that there was big bang, but only a bounce: And the Black Hole entropy calculation, which is nice, but not entirely satisfactory yet.
Does this describe nature? We do not know...
carlo rovelli
Best Answer
Here is something, which may be aiming a little low...
The main two ways we describe our universe, quantum mechanics and general relativity, contradict each other when applied simultaneously. This seems to point out that the quantum nature of spacetime itself needs to be understood better.
One way to resolve theoretical problems with our current understanding of spacetime is to embed it in a larger theoretical structure, which has powerful underlying symmetries. Those symmetries cannot be too restrictive: they should be enough not only to make the model well-behaved, but also to be consistent with the not-quite-so-symmetric world we see around us.
This way of thinking has led to the theoretical structure of string theory and then M-theory. To study the structure of the theory, it is useful to first concentrate on the most symmetric situations, even though these are the most removed from our world. At first, this led to study of supersymmetric string theories in 10 dimensions (higher dimensional theories are more symmetric - their Lorentz invariance is larger and more restrictive). Later it turned out that those are all secretly related to an even larger and more symmetric structure, dubbed M-theory, which describes all the previously known string theories as well as 11-dimensional supergravity.
The story is not finished, we only have bits and pieces of the underlying structure that is M-theory. But, we do have many indications we are on the right track. As always with deep structures we found side benefits in the form of unexpected applications in mathematics and physics.
One of the unexpected discoveries is that quantum gravity is not all that different from other parts of physics, and sometimes conventional physics can be reformulated in different variables to make it equivalent to a quantum gravitational theory. Using classical and semi-classical gravity calculations then helps us explore conventional physics in regimes otherwise inaccessible. This is the whole subject of holography and its applications.
So, what we seem to have found is, instead of a specific model to describe our universe at short distances (or high energies), a whole new language in which we describe and discuss physics - and not just high energy physics. Where precisely this is going to lead is anyone’s guess.
Now, if your grandmother feels patronized, has more knowledge of physics and would like to ask more specific questions, I can try to add more details. It is a very large subject...
(see also the answers to this similar question)