Modules over monoids vs algebra over monads
Solution 1:
Monoid actions for a fixed monoid $M$ are an instance of being algebras for a monad.
Specifically, let the monad be $T:\mathcal{Set}\to\mathcal{Set},\ X\mapsto M\times X$ (which is, by the way, the underlying set of the free $M$-module generated by $X$), with monad multiplication induced by the monoid multiplication: $X\times M\times M\to X\times M$ and monad unit induced by the monoid unit $X\to X\times M,\ x\mapsto (x,1_M)$.