What is the heuristic idea behind the FourierMukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?
What is the heuristic idea behind the FourierMukai transform? What is the connection to the classical Fourier transform? Moreover, could someone recommend a concise introduction to the subject? 


First, recall the classical Fourier transform. It's something like this: Take a function $f(x)$, and then the Fourier transform is the function $g(y) := \int f(x)e^{2\pi i xy} dx$. I really know almost nothing about the classical Fourier transform, but one of the main points is that the Fourier transform is supposed to be an invertible operation. The FourierMukai transform in algebraic geometry gets its name because it at least superficially resembles the classical Fourier transform. (And of course because it was studied by Mukai.) Let me give a rough picture of the FourierMukai transform and how it resembles the classical situation.
But to make all of this actually work out, we have to actually use the derived pushforward, not just the pushforward. And so we have to work with the derived categories. When $X$ is an abelian variety, $Y$ is the dual abelian variety, and $\mathcal{P}$ is the socalled Poincare line bundle on $X \times Y$, then the FourierMukai transform gives an equivalence of the derived category of coherent sheaves on $X$ with the derived category of coherent sheaves on $Y$. I think this was proven by Mukai. I think this is supposed to be analogous to the statement I made about the classical Fourier transform being invertible. In other words I think the Poincare line bundle is really supposed to be analogous to the function $e^{2\pi i xy}$. A more general choice of $\mathcal{P}$ corresponds to, in the classical situation, socalled integral transforms, which have been previously discussed here. This is probably why $\mathcal{P}$ is called the integral kernel. You may also be interested in reading about Pontryagin duality, which is a version of the Fourier transform for locally compact abelian topological groups  this is obviously quite similar, at least superficially, to Mukai's result about abelian varieties. However I don't know enough to say anything more than that. There are some cool theorems of Orlov, I forget the precise statements (but you can probably easily find them in any of the books suggested so far), which say that in certain cases any derived equivalence is induced by a FourierMukai transform. Note that the converse is not true: some random FourierMukai transform (i.e. some random choice of the sheaf $\mathcal{P}$) is probably not a derived equivalence. I think Huybrechts' book "FourierMukai transforms in algebraic geometry" is a good book to look at. Edit: I hope this gives you a better idea of what is going on, though I have to admit that I don't know of any good heuristic idea behind, e.g., Mukai's result  it is analogous to the Fourier transform and to Pontryagin duality, and thus I suppose we can apply whatever heuristic ideas we have about the Fourier transform to the FourierMukai transform  but I don't know of any heuristic ideas that explain the FourierMukai transform in a direct way, without appealing to any analogies to things that are outside of algebraic geometry proper. Hopefully somebody else can say something about that. But  there is certainly something deep going on. Just as CommRing behaves a lot like Set^{op}, I think there is probably some kind of general phenomenon that sheaves (or vector bundles) behave a lot like functions, which is what's happening here. Pullback of sheaves behave a lot like pullback of functions... Pushforward of sheaves behave a lot like integration of functions... Tensor product of sheaves behave a lot like multiplication of functions... 


You may want to look at Tom Bridgeland's PhD thesis. 


Just a complement to the answer of Kevin Lin. There is a case where the analogy between sheaves and functions is more than analogy : the case of varieties over finite field. More precisely, if X is a variety over F_p and F is a ladic constructible sheaf on X, one can associate to F a function (in a set theoretic sense) over the set of F_p points of X by mapping x to the trace of the Frobenius acting on the fiber of F at x. This defines a correspondace sheaffunction compatible with all the analogies cited by Kevin. If we fix a character of F_p then one have the usual Fourier transform for functions over F_p. One can ask for an analogue for the ladic sheaves over the affine space A. It exists, it is the FourierDeligne transform. The fact that the function associated to the FourierDeligne transform of a sheaf is the (usual) Fourier transform of the function associated to the sheaf is a consequence of the Grothendieck trace formula. In fact, the FourierDeligne transform is a FourierMukai transform for the derived category of ladic constructible sheaf on A ! Ok, when one speaks about FourierMukai, one think about complex algebraic geometric and categories of coherent sheaves but I think that to have the above situation where we have a really sheaf/function dictionnary in mind can be useful. This dictionnary was one of the motivation for the formulation of the geometric Langlands program (see some expository articles of Frenkel for example). 


The following answers might be useful:
The last one has my sketch of an answer which I'll post here once it gets better. 


I second Kevin's suggestion of Huybrechts' book, but if you want to to look at something shorter first I recommend the notes by Hille and van den Bergh. 


Alexander Polishchuk, Abelian Varieties, Theta Functions and the Fourier Transform, Cambridge Tracts in Mathematics 153, Cambridge University Press, 2003. This also happens to be one of my favourite books. 

