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