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.