One standard example of a monoid is the monoid $\mathbb N$ of natural numbers. The monoid ring $\mathbb C[\mathbb N]$ is equal to the polynomial ring $\mathbb C[T]$; the study of this ring and its modules (which are just representations of $\mathbb N$) is a pretty standard topic in algebra. More generally, if the ideal $I \subset \mathbb C[T_1,\ldots,T_n]$ is the generated by differences of pairs of monomials $m_i(\underline{T}) - m_i'(\underline{T})$ ($i = 1,\ldots,s$), then the quotient $\mathbb C[T_1,\ldots,T_n]/I$ is equal to the monoid ring of the associated commutative monoid $$\langle T_1,\ldots, T_n \, | \, m_i(\underline{T}) = m_i'(\underline{T}) (i = 1,\ldots,s)\rangle.$$ (In his answer, Qiaochu gives the example of $\mathbb C[T]/(T^2-T^3)$.) These kind of rings come up quite a bit in the study of toric geometry and log geometry; they are particularly nice examples of affine rings. Representations of the corresponding monoids are just modules over these rings.
However, when people study these rings and their modules, they tend to use the language of commutative algebra and algebraic geometry (and also quite a lot of combinatorial language, in the context of toric geometry). For the reasons noted in other answers, there doesn't seem to be that much advantage to using a more representation-theoretic viewpoint, because these rings are less special (among all rings) then group rings are.