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).