Relations between monoids and modules? What is the relation between monoids and modules? Are they completely different algebraic structures, or is there a kind of inclusion relation like "elements of a module are also elements of a monoid"? 
 A: A module is an abelian group.  (It's more useful to think of a module as the analog of a vector space, but with the set of scalars coming from a ring instead of a field.  Usually, one arrives at this notion of a module in terms of "the action of a ring on a set" where the set is a module.)
A monoid is a relaxation of the definition of a group.  A monoid has an associative operation and a neutral element, but makes no promises about inverses.
I don't see how to express any more of a relation than "all modules are monoids" but only for the dull reason that all (abelian) groups are (abelian) monoids with the added constraint that every element has an inverse.
A: There is a chain  of forgetful functors which progressively forgets the various operations in the structure: $$\mathrm{Mod_R}\to\mathrm{Ab}\to\mathrm{AbMon}\to\mathrm{Set}$$
The interesting thing is that you can go in the opposite direction too with free functors $$\mathrm{Set}\to\mathrm{AbMon}\to\mathrm{Ab}\to\mathrm{Mod_R}$$ 
Each forgetful functor $U$ is adjoint to the respective free functor $F$
