TITLE: What is the general geometric interpretation of modules in algebraic geometry? 
QUESTION [35 upvotes]: Algebraic geometry is quite new for me, so this question may be too naive. therefore, I will also be happy to get answers explaining why this is a bad question.
I understand that the basic philosophy begins with considering an abstract commutative ring as a function space of a certain "geometric" object (the spectrum of the ring). I also understand that at least certain types of modules correspond to well known geometric constructions. For example, projective modules should be thought of as vector bundles over the spectrum (and that there are formal statements such as the Serre-Swan theorem which make this correspondence precise in certain categories). My question is, what is the general geometric counterpart of modules? 
This is not a formal mathematical question, and I am not looking for the formal scheme-theoretic concept (of sheaves of certain type and so on), but for the geometric picture that I should keep in mind when working with modules.
I will appreciate any kind of insight or even just a particularly Enlightening example.

REPLY [6 votes]: These somewhat naive remarks are motivated by the desire to think about the question myself.
Sections of bundles are natural examples of (locally) free modules and the standard examples are the tangent and cotangent bundles of smooth varieties.  The need for more general modules is the fact that kernels and cokernels of maps of (locally) free modules need not be so.  In addition to ideal sheaves of subvarieties, as noted above the kahler differentials are a standard example of not necessarily locally free sheaves.  
These two remarks are connected by the standard exact sequence of a closed subscheme, which exhibits the sheaf of kahler differentials of a local complete intersection, as the cokernel of a map of locally free sheaves.  I,.e, the kahler differentials on a closed subvariety of a smooth variety, are the quotient of the kahler differentials on the ambient variety by the conormal sheaf of the subvariety.
Thus, the standard examples of modules are ideals of subvarieties and kahler differentials, and at least for lci subvarieties these arise as either kernels or cokernels of maps of bundles.  So geometrically one could say one wants to consider differentials and subschemes, and algebraically one wants to be able to take kernels and cokernels.