2 families of examples that are sometimes useful to have in mind:

(1)  The group ring of a non-abelian finite group over a finite commutative ring.

and 

(2)  the incidence algebra of a finite poset over a finite commutative ring (the ring of upper triangular matrices is a basic example of this).  

Of course, both of these are special cases of the same more general categorical (or quiver) definition.  Before I wrote that I'd never dealt with the more general concept, but that was a lie...