What is a fuzzy space? Can someone give a down-to-earth explanation of what is a fuzzy space? (As known from M-theory and noncommutative geometry)
 A: Fuzzy spaces can refer to a few things, but mainly they refer to finite-dimensional spectral triples, specifically noncommutative ones. 
So to understand the idea behind them it is useful to understand what a spectral triple is (roughly). Then to look at noncommutative versions. 
So a spectral triple is a collection of algebraic objects, $(A, H, D)$, which characterise a geometry completely. To see that this is possible is not an easy exercise, but has been shown to hold for compact Riemannian spin manifolds by Alain Connes and the work surrounding it. So we take such a manifold $(M, g, S)$ where $S$ is a spinor bundle over $M$. Then we can set the object $A$ to be the algebra of smooth functions over $M$, $C^\infty(M)$ which is always commutative (there are some technicalities about $A$ here but this is the rough idea), we take $H$ to be the Hilbert space of square-integrable spinor sections, usually denoted $L^s(\Gamma(S))$ and $D$ is given by the Dirac operator. 
The key thing to note is the conditions on $(A, H, D)$ to make the connection between this algebraic data and the geometrical data $(M, g, S)$ do not depend on the commutativity of the algebra $A$. So the idea of noncommutative geometry is to generalise the algebraic side of this bridge to when $A$ is noncommutative. And to see what happens and whether we can make sense of anything. And a lot has been done in this regard, (see the standard model and a noncommutative geometry, noncommutative spacetimes, quantum groups etc). 
Now fuzzy spaces are just a very specific case, when we take the algebra $A$ to be a matrix algebra or a finite direct sum of matrix algebras, i.e. a finite-dimensional noncommutative algebra. 
So $A= M_n(\mathbb{C})$ for some value of $n$, and then we look at a Hilbert space on which we have a representation of $A$, and then we look at a specific operator on this Hilbert space (the operator has to satisfy the same list of conditions as it does for the manifold spectral triple above). 
Understanding what you have found is not so easy, but with the help of Lie group/algebra actions and quantum group actions, we can start to examine the symmetries of these spaces. 
Now string theory and particle physics in general, take a combination of a manifold spectral triple $(C^\infty, L^2(\Gamma(S)), S)$ and some finite noncommutative spectral triple (i.e. a fuzzy space) and then look at what you end up with. 
You could think of this is a Kaluza-Klein idea, but now instead of attaching a circle or compact manifold to every point, we attach a fuzzy space to every point. These are called almost-commutative spacetimes. 
Why do we do this? It turns out a lot of issues with the normal Kaluza-Klein method appear to get resolved. However, a lot of other problems arise and active research is being undertaken into whether or not the idea of almost-commutative spacetimes is a good one, and whether or not it can be improved. 
There are constructions of fuzzy spheres and other generalisations of simple manifolds, by looking at the symmetries of the spectral triples and them looking at how that spectra (eigenvalues) of the Dirac operator $D$ compare with the commutative (normal) manifold spaces. 
Hopefully this helps. 
