Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative monoid?

Subsemigroups of finite commutative (even cyclic) semigroups are complicated. For example every finite lattice embeds into the lattice of subsemigroups of a finite cyclic semigroup (hence into the lattice of submonoids of one of the $\mathbb{Z}_n$. See Repnitskii, Vladimir, On subsemigroup lattices without nontrivial identities. Algebra Universalis 31 (1994), no. 2, 256–265. 


To supplement Mark's answer, note that every noninvertible element of $Z/p^n$ is nilpotent. So the idempotents in this case are 0,1. Thus if m is divisible by exactly k primes, then $Z/m$ has semilattice of idempotents the power set of a kelement set by the Chinese remainder theorem. Of course every k element semilattice embeds in the power set of a kelement set so even the lattice of semilattice submonoids can be complicated. 

