Recall that the default interpretation of the word "module" is a unitary module. If M contains x, and 1x = 0, then x in S cannot faithfully map to x in M. Multiply by 1, and x maps to 0 instead of x. So we assume M is unitary, whence F is a free object in the category of unitary modules.
If S is finite then F and M are finitely generated. If the kernel of the homomorphism from F onto M is also finitely generated, then M is finitely presented. Each generator of the kernel is called a relation. Thus a finitely presented module can be described using finitely many generators and relations.
Let's take a simple example. Let R be the ring of integers. Let 1 and q act as generators for a free module over R. At this point the module is Z*Z. Bring in one relation, 7q-2 = 0. Now perform algebra as usual, but replace each instance of 7q with 2 as you go. The module is infinite, but it is finitely presented, with two generators and one relation. Though not obvious at first, it is isomorphic to Z, generated by 4q-1.
When a module is finitely generated, with n generators, endomorphisms are defined by n by n matrices over R that map each generator to a linear combination of said generators. If the module M is free then matrices and endomorphisms correspond 1-1. If M is not free then a nontrivial linear combination of generators is equal to 0. Write this linear combination into the row of a matrix, or put in a row of zeros; the result is the same. In any case, matrix addition and multiplication correspond to addition and multiplication in the ring of endomorphisms.