A group has lots of interesting characteristic subgroups (center, commutator and derived subgroups, Frattini subgroup, Fitting subgroup, the identity component for topological groups, the torsion subgroup for abelian groups). Often, these subgroups play an important role in the classification of the groups.
But what about characteristic subrings? What are interesting examples? I am especially interested in commutative rings, so that the center doesn't really qualify. The only example I know at the moment is the prime ring (the subring generated by the unit).
Since characteristic subgroups are normal and normal subgroups in group theory correspond to ideals in ring theory, perhaps a better question would be: What are interesting characteristic ideals of a (commutative) ring? Here only the nil radical and the Jacobson radical come into my mind.