This more of a generalization than a direct answer, but it may prove interesting and/or useful to the original poster.
Universal Algebra has the student look at the lattice of congruences of various algebraic structures. Simple algebras are then algebras with a lattice of two congruences, i.e. no nontrivial congruences, which for groups corresponds to no nontrivial normal subgroups. A related concept is that of subdirectly irreducible algebra. Here the congruence lattice has a unique nontrivial smallest congruence, that is a congruence which is contained in any other congruence on the algebra (except the trivial one induced by an isomorphism). Any simple algebra is subdirectly irreducible. The utility of the latter concept is that any algebra has a representation as a subdirect product (subalgebra of a direct product) of subdirectly irreducible algebras. So when one looks at classes of algebras (of a single similarity type) which are closed under taking direct products and subalgebras (and often isomorphic images of such), one finds the subdirectly irreducible algebras as natural building blocks to form the class.
I recall that semilattices and Boolean algebras had nice classifications of finite subdirectly irreducible algebras. I am confident the general algebra literature contains more.
Gerhard "Ask Me About System Design" Paseman, 2011.04.03