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 definition; I've never personally used any other example.

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...