TITLE: Classifications of finite simple objects
QUESTION [7 upvotes]: I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "classification" in the informal sense of the term, but also answers that take into account a more sophisticated viewpoint (as in this mo question) are welcome, as well as answers that consider reasonable weaker notions of classification (as e.g. this mo question, that asks about a classification of finite simple groups up to finitely many exceptions).
The (apperarently incomplete) case of finite commutative rings has already been discussed here. The finite p-groups have been considered here.
Also answers/remarks involving the classification of "finite simple objects" of some category (or higher category) are considered in topic (provided that a reasonable definition of "finite" and "simple" is suggested in that context).

REPLY [3 votes]: Somewhat related to Igor Pak's comment is the classification of the finite irreducible Coxeter groups. Of course they are not "simple" as groups, but the irreducibility seems the natural replacement for simplicity; here "irreducible" means that the Coxeter diagram is connected, or equivalently, that the Coxeter system does not split as the direct product of two Coxeter systems.
The outcome is the famous list $A_n$, $B_n = C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, $G_2$, $H_3$, $H_4$, $I_{(n)}$, where the last three items are maybe less well known people only familiar with Lie groups and Lie algebras and/or algebraic groups since they don't survive there.
See also http://en.wikipedia.org/wiki/Coxeter_group.