Quivers can be important in AG:

Suppose you have a finite subgroup G of SL(n,C)

Then G acts on C^n, and you can consider the (GIT) quotient Cⁿ /G. Now this space is usually very singular.

To G you can associate a quiver, whose nodes correspond to irreducible representations of G. You can also impose certain relations on the path algebra of this quiver, to get a quiver with relations (Q,R) associated to the finite group G.

Now we can consider the notation of quiver representations. In particular, we can consider the "moduli space" M(Q,R) of all (semi-stable) representations of the quiver (Q,R) associated to G.

Then (a connected component of) M(Q,R) is a smooth variety, which is a resolution of singularities of C^n/G.