We study irreducible characters, and define and study supercharacters of the adjoint group of a finite radical ring. We extend work of Halasi on irreducible characters, and recent work of Diaconis and Isaacs on supercharacters of algebra groups defined over finite fields, and derive some consequences for the representation theory of compact p-adic algebra groups. We also define the notion of a fully ramified supercharacter, and consider orthogonal decompositions of a given supercharacter. Finally, we obtain a necessary and sufficient condition for a supercharacter to have a unique irreducible constituent, and illustrate the result in the case of the unitriangular group over a finite field.