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):
Also, many information on cluster algebras (the definition of which requires quiver mutation) can be found at the cluster algebra portal
Some very nice introductions and surveys to some of the theories which were developped thanks to cluster algebras and mutation are:
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;
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: