  The aim of this article is to study the ideal class monoid
$\mathcal{C}\ell(S)$ of a numerical semigroup $S$ introduced by V. Barucci and
F. Khouja. We prove new bounds on the cardinality of $\mathcal{C}\ell(S)$. We
observe that $\mathcal{C}\ell(S)$ is isomorphic to the monoid of ideals of $S$
whose smallest element is 0, which helps to relate $\mathcal{C}\ell(S)$ to the
Ap\'ery sets and the Kunz coordinates of $S$. We study some combinatorial and
algebraic properties of $\mathcal{C}\ell(S)$, including the reduction number of
ideals, and the Hasse diagrams of $\mathcal{C}\ell(S)$ with respect to
inclusion and addition. From these diagrams we can recover some notable
invariants of the semigroup. Lastly, we prove some results about irreducible
elements, atoms, quarks and primes of $(\mathcal{C}\ell(S),+)$. Idempotent
ideals coincide with over-semigroups and idempotent quarks correspond to
unitary extensions of the semigroup. We show that a numerical semigroup is
irreducible if and only if $\mathcal{C}\ell(S)$ has at most two quarks.
