Hello to all,
The phrase "quiver mutation has been invented by Fomin and Zelevinsky and has found numerous applications throughout mathematics and physics" is one that some of us encountered on a near-daily basis. I personally know only of a few applications. I thus propose the following thread: how have you used quiver mutation in your area of expertise ?
The work of Fomin and Zelevinsky is very beautiful and important (BTW I can recommend Zelevinsky's GRASP lecture for video and notes on related subjects) but the theory of quiver mutations goes back much earlier, at least to the work of Bernstein, Gelfand and Ponomarev on reflection functors for representations of quivers (which give derived equivalences of their categories of representations). This (together with Beilinson's description of the derived category of projective space and the Koszul duality of Bernstein Gelfand Gelfand) is one of the origins of an entire school (Bondal, Rudakov, Gorodentsev, others) studying exceptional collections (roughly, describing derived categories of varieties as quiver representations) and the operation of mutation, taking one exceptional collection to another. This subject has been revived due in part to work in homological mirror symmetry (Kontsevich, Seidel, Bondal, Orlov, Auroux, Katzarkov....) where the mutations are mirror to Dehn twists for symplectic Lefschetz fibrations. There are also close relations of mutations with their "Calabi-Yau brethren", the spherical twists functors, of Seidel-Thomas.
Another source of quiver mutations comes from Seiberg duality of N=1 gauge theories in four dimensions -- these gauge theories are prescribed by quivers with potential, and mutations induce dualities of the gauge theories. (This is related to the previous story via AdS/CFT I believe, but I'm just misquoting Aaron Bergman here so you can look at his papers for great explanations of this for mathematicians).
Of course there are many other directions to mention, but these are all (historically) independent of the wonderful work on cluster algebras of Fomin-Zelevinsky and the theory of mutations for quivers with potential, as far as I know -- though of course mathematically everything's actually the same, and the relation between cluster algebras and mirror symmetry and gauge theories have emerged in the last few years (see eg Kontsevich-Soibelman and Gaiotto-Moore-Neitzke) -- and there are many people on MO better qualified to comment on all of this than me, sorry for any offensive anachronisms/omissions/..
First of all, I would like to note that there is a very nice applet, due to Keller, which mutates quivers (and does much more): http://people.math.jussieu.fr/~keller/quivermutation/
Also, many information on cluster algebras (the definition of which requires quiver mutation) can be found at the cluster algebra portal http://www.math.lsa.umich.edu/~fomin/cluster.html
Some very nice introductions and surveys to some of the theories which were developped thanks to cluster algebras and mutation are:
http://people.math.jussieu.fr/~keller/publ/KellerCatAcyclic.pdf
http://people.math.jussieu.fr/~keller/publ/KellerClusterAlgQuivRep.pdf
http://uk.arxiv.org/pdf/1012.4949.pdf
http://uk.arxiv.org/pdf/1012.6014.pdf
http://people.math.jussieu.fr/~keller/publ/KellerCYtriangCat.pdf
Here are a few examples of areas of research which are related to (or motivated by) Fomin-Zelevinsky's quiver mutation:
Cluster tilting theory (in representation theory of quivers and algebras);
Triangulations of punctured Riemann surfaces;
Higher Teichmuller spaces;
Poisson geometry;
In algebraic geometry: Stability conditions, Calabi-Yau algebras, Donaldson-Thomas invariants...
Let me give a few more details on cluster tilting: The definition of a cluster algebra makes use of the notion of seed mutation. Quiver mutation is a part of this seed mutation. As an analogy, one can consider the flip of triangulations of an n-gone (To flip a triangulation, delete one of its arcs and replace it by the only arc giving a new triangulation). Through this analogy, seeds correspond to triangulations, and seed mutation to flips.
Now, in the representation theory of finite dimensional algebras, there is a notion of tilting modules. Such modules can sometimes be mutated at an indecomposable summand (as triangulations can be flipped at an arc), but not always: somme summands cannot be mutated. Moreover, there is a quiver naturally associated with such a module (the Gabriel quiver of its endomorphism algebra). Through a mutation, the associated quivers are related by Fomin-Zelevinsky's quivers mutation in some cases, but not always.
The whole theory of cluster tilting, including cluster categories and their generalisations, module categories over preprojective algebras, more general Calabi-Yau triangulated categories... arised from the (successful) attempt to fix these two problems in the relation between tilting theory and cluster algebras.
As a concrete application of this theory, on can cite Keller's proof of Zamolodchikov's periodicity conjecture:
http://people.math.jussieu.fr/~keller/publ/KellerPeriodicity.pdf
To the best of my understanding an exceptional collection of elements in $\mathcal{D}(X)$ determines a quiver - and quiver mutations come to represent corresponding mutations among the elements of the collection, as developed by Rudakov et. al. (vertices of the quiver). In this context, there is a beautiful construction due to P. Seidel - on the mirror analog of such mutations in the Fukaya category
