free module
A free module is the module that behaves most like a vector space: it has a basis. Every element is a unique finite combination of basis elements with coefficients in the ring, so you can do honest coordinate computations exactly as in linear algebra. Free modules are the simplest, most transparent modules, and every module is a quotient of one — they are the raw material from which all modules are built.
Formally, an R-module F is free on a set B (a subset of F) if every element of F is a unique R-linear combination of finitely many elements of B; equivalently, F is isomorphic to a direct sum of copies of R, one copy per element of B. The cardinality of B is the rank. The defining universal property: any function from B to an R-module M extends uniquely to an R-module homomorphism F -> M, so free modules are the free objects in the category of R-modules.
Over a field every module is free (that is exactly the existence of a basis for any vector space), but over a general ring most modules are not free — Z/2Z is not a free Z-module, since no nonzero subset is linearly independent. A subtle point: rank is well defined for commutative rings, but there exist (non-commutative) rings R with R isomorphic to R-squared as modules, so even the rank can misbehave outside the commutative world.
R^n with standard basis e_1, ..., e_n is free of rank n over R. Over Z, the module Z ⊕ Z is free of rank 2 with basis (1,0), (0,1), while Z/2Z, having no Z-linearly independent element, is not free at all.
Direct sums of copies of R are the free modules; torsion modules like Z/2Z never are.
Every free module is projective and flat, but the converses fail; over a PID, however, every submodule of a free module is free, which is the technical heart of the structure theorem. Over a local ring, finitely generated projective implies free (this is a form of Nakayama's lemma in action).