TITLE: What are your favorite finite non-commutative rings?
QUESTION [19 upvotes]: When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand. 
There are many convenient examples of commutative rings, both finite and infinite, and there are many convenient examples of infinite non-commutative rings. But I don't have a good collection of finite non-commutative rings to think about. I usually just think of a matrix ring over a finite field.
Do YOU have other examples that you particularly like/find easy to use/find to be a good source of counterexamples?

REPLY [5 votes]: The path algebra of an acyclic quiver (no loops or closed circuits) over a finite field. These rings are hereditary, i.e. they have projective dimension $\leq 1$ and occupy an intermediate position between Greg's semisimple and wild examples. If a quiver is "arboreal" (underlying undirected multigraph is in fact a tree) then its path algebra may be identified with the incidence algebra of the poset of vertices in which $i\leq j$ $\iff$ there is a directed path from $i$ to $j$ (by the assumption, there is at most one).
For any ring $R$ you can form matrix algebra $M_n(R)$ and they will be Morita equivalent. In particular, matrix rings of path algebras are also hereditary, but their simple modules are not all one-dimensional.
By the way, I think that in the noncommutative setting, when Frobenius morphism is not available, this is a wrong question: all finite-dimensional algebras over a field should be considered on a similar footing, regardless of whether the field is finite or infinite.