Give a description of Loop Quantum Gravity your grandmother could understand Of course, assuming your grandmother is not a theoretical physicist.
I'd like to hear the basics concepts that make LQG tick and the way it relates to the GR. I heard about spin-networks where one assigns Lie groups representations to the edges and intertwining operators to the nodes of the graph but at the moment I have no idea why this concept should be useful (except for a possible similarity with gauge theories and Wilson loops; but I guess this is purely accidental). I also heard that this spin-graph can evolve by means of a spin-foam which, I guess, should be a generalization of a graph to the simplicial complexes but that's where my knowledge ends.
I have also read the wikipedia article but I don't find it very enlightening. It gives some motivation for quantizing gravity and lists some problems of LQG but (unless I am blind) it never says what LQG actually is.
So, my questions:

  
*
  
*Try to give a simple description of fundamentals of Loop Quantum Gravity.
  
*Give some basic results of the theory. Not necessary physical, I just want to know what are implications of the fundamentals I ask for in 1.
  
*Why is this theory interesting physically? In particular, what does it tell us about General Relativity (both about the way it is quantized and the way it is recovered from LQG).
  

 A: 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 
A: @Marek your question is very broad. Replace "lqg" with "string theory" and you can imagine that the answer would be too long to fit here ;>). So if this answer seems short on details, I hope you will understand.
The program of Loop Quantum Gravity is as follows:


*

*The notion of diffeomorphism invariance background independence, which is central to General Relativity, is considered sacrosanct. In other words this rules out the String Theory based approaches where the target manifold, in which the string is embedded, is generically taken to be flat [Please correct me if I'm wrong.] I'm sure that that is not the only background geometry that has been looked at, but the point is that String Theory is not written in a manifestly background independent manner. LQG aims to fill this gap.

*The usual quantization of LQG begins with Dirac's recipe for quantizing systems with constraints. This is because General Relativity is a theory whose Hamiltonian density ($\mathcal{H}_{eh}$), obtained after performing a $3+1$ split of the Einstein-Hilbert action via the ADM procedure [1,2], is composed only of constraints, i.e.
$$ \mathcal{H}_{eh} = N^a \mathcal{V}_a + N \mathcal{H} $$
where $N^a$ and $N$ are the lapse and shift vectors respectively which determine the choice of foliation for the $3+1$ split. $\mathcal{V}_a$ and $\mathcal{H}$ are referred to as the vector (or diffeomorphism) constraint and the scalar (or "hamiltonian") constraint. In the resulting phase space the configuration and momentum variables are identified with the intrinsic metric ($h_{ab}$) of our 3-manifold $M$ and its extrinsic curvature ($k_{ab}$) w.r.t its embedding in the full $3+1$ spacetime, i.e.
$$ {p,q} \rightarrow \{\pi_{ab},q^{ab}\} := \{k_{ab},h^{ab}\} $$
This procedure is generally referred to as canonical quantization. It can also be shown that $ k_{ab} = \mathcal{L}_t h_{ab} $, where $ \mathcal{L}_t $ is the Lie derivative along the time-like vector normal to $M$. This is just a fancy way of saying that $ k_{ab} = \dot{h}_{ab} $
This is where, in olden days, our progress would come to a halt, because after applying the ADM procedure to the usual EH form of the action, the resulting constraints are complicated non-polynomial expressions in terms of the co-ordinates and momenta. There was little progress in this line until in 1986 $\sim$ 88, Abhay Ashtekar put forth a form of General Relativity where the phase space variables were a canonically transformed version of $ \{k_{ab},h^{ab}\} $ This change is facilitated by writing GR in terms of connection and vielbien (tetrads) $ \{A_{a}^i,e^{a}_i\}$ where $a,b,\cdots$ are our usual spacetime indices and $i,j,\cdots$ take values in a Lie Algebra. The resulting connection is referred to as the "Ashtekar" or sometimes "Ashtekar-Barbero" connection. The metric is given in terms of the tetrad by :
$$ h_{ab} = e_a^i e_b^j \eta_{ij} $$
where $\eta_{ij}$ is the Minkowski metric $\textrm{diag}(-1,+1,+1,+1)$. After jumping through lots of hoops we obtain a form for the constraints which is polynomial in the co-ordinates and momenta and thus amenable to usual methods of quantization:
$$ \mathcal{H}_{eha} = N^a_i \mathcal{V}_a^i + N \mathcal{H} + T^i \mathcal{G_i} $$
where, once again, $ \mathcal{V}_a^i $ and $\mathcal{H}$ are the vector and scalar constraints. The explanation of the new, third term is postponed for now.
Nb: Thus far we have made no modifications to the theoretical structure of GR. The Ashtekar formalism describes the exact same physics as the ADM version. However, the ARS (Ashtekar-Rovelli-Smolin) framework exposes a new symmetry of the metric. The introduction of spinors in quantum mechanics (and the corresponding Dirac equation) allows us to express a scalar field $\phi(x)$ as the "square" of a spinor $ \phi = \Psi^i \Psi_i $. In a similar manner the use of the vierbien allows us to write the metric as a square $ g_{ab} = e_a^i e_b^j \eta_{ij} $. The transition from the metric to connection variables in GR is analogous to the transition from the Klein-Gordon to the Dirac equation in field theory.

*The application of the Dirac quantization procedure for constrained systems shows us that the kinematical Hilbert space, consisting of those states which are annihilated by the quantum version of the constraints, has spin-networks as its elements. All of this is very rigorous and several mathematical technicalities have gradually been resolved over the past two decades.

This answer is already pretty long. It only gives you a taste of things to come. The explanation of the Dirac quantization procedure and spin-networks would be separate answers in themselves. One can give an algorithm for this approach:


*

*Write GR in connection and tetrad variables (in first order form).

*Perform $3+1$ decomposition to obtain the Einstein-Hilbert-Ashtekar Hamiltonian $\mathcal{H}_{eha}$ which turns out to be a sum of constraints. Therefore, the action of the quantized version of this Hamiltonian on elements of the physical space of states yields $ \mathcal{H}_{eha} \mid \Psi \rangle = 0 $. (After a great deal of investigation) we find that these states are represented by graphs whose edges are labeled by representations of the gauge group (for GR this is $SU(2)$).

*Spin-foams correspond to histories which connect two spin-networks states. On a given spin-network one can perform certain operations on edges and vertices which leave the state in the kinematical Hilbert space. These involve moves which split or join edges and vertices and those which change the connectivity (as in the "star-triangle transformation"). One can formally view a spin-foam as a succession of states $\{ \mid \Psi(t_i) \rangle \}$ obtained by the repeated action of the scalar constraint $ \mid \Psi(t_1) \rangle \sim \exp{}^{-i\mathcal{H}_{eha}\delta t} \mid \Psi (t_0); \mid \Psi(t_2) \rangle \sim \exp{}^{-i\mathcal{H}_{eha}\delta t} \mid \Psi (t_1) \cdots \rangle $ [3].

*The graviton propogator has a robust quantum version in these models. Its long-distance limit yields the $1/r^2$ behavior expected for gravity and an effective coarse-grained action given by the usual one consisting of the Ricci scalar plus terms containing quantum corrections.
There is a great deal of literature to back up everything I've said here, but this is already pretty exhausting so you'll have to take me on my word. Let me know what your Grandma thinks of this answer ;).

A: The answers above already cover the necessary ground, but they did not mention one 'angle' of it that i find particularly illuminating: holonomy-flux algebras.
The Ashtekar variables describing the metric ($SU(2)$-valued decomposition of the metric) can be understood in terms of its holonomies — and, much like it's done in gauge theories, one can deal with these holonomies' algebras.
I find this interesting because it seems to link to Gauge Theories in a very particular way, à la Wilson Loops. And, if you squint a bit, ;-) , you can also think of 'flux compactification' (string theory; cf. Green-Schwarz-Witten, Vol 2, chapter 14, or http://arxiv.org/abs/hep-ph/0107039) as a much broader 'object' in Physics.
Just my 2¢.
A: Let me make an attempt: LQG is a name for a collection of research programs aimed at quantizing a metric theory of gravity (usually the Einstein-Hilbert action) directly. Those attempts use different variables (starting with Ashtekar's new variables) and different techniques of quantization (e.g path integrals, canonical quantization, loop quantization, etc.) in an attempt to circumvent the problems one encounters when perturbatively quantizing the EH action around flat spacetime.
For a system with finitely many degrees of freedom, unless you are making some preverse choices, quantizing a classical theory usually gives a quantum theory whose classical limit, in turn, is the theory you started from. Also, most the above quantization procedures (with the possible exception of loop quantization) usually give the same quantum theory, perhaps differing in some minor ways. 
All of this is far from guaranteed in field theories, because of ultra violet divergences. The questions of whether when you "quantize GR" the resulting theory has a classical limit where it turns into classical GR in flat spacetime, or whether all the different quantizations used in LQG are equivalent to each other, those are open questions as far as I can tell.
The names of different fields tend to be historical, in this case it has to do with the loop variables employed in constructing the so-called kinematical Hilbert space of LQG. Non -perturbative quantum gravity (or simply quantum gravity) usually refers to more than just LQG, for example including things like causal dynamical triangulations (CDT). All of those approaches have in common the belief that some appropriate approach to quantization will enable us to quantize the metric (or some alias thereof) directly. 
This is in distinction to the approach titled (for similarly obscure historical reasons) "string theory" in which the fundamental degrees of freedom are not the metric (or the connection, or anything else appearing in the low energy description of gravity), and the low energy degrees of freedom are "emergent" in some appropriate sense of that word.
