Vector spaces are completely characterized by their dimension (up to an isomorphism). In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules. However, every module is a cokernel of a homomorphism of free modules.
veCtor SPaCes aRE coMplETely cHarACtERizEd by theIr dimEnsion (up to An IsomorPhism). in gEnErAL, tHErE is Not SuCh A cOmpLeTe claSsificatiOn for mOdules, eVen If one restriCts OneSELf TO FIniTeLy GenErAted moduLes. HoweveR, EverY moDulE IS a Cokernel Of a homomorphiSm of frEe modUles.