Roughly a module can be thought of as a vector bundle on the spectrum, where the dimension of fibers may vary. Let me give some examples and facts:

 - A free modules corresponds to trival vector bundles, or more generally projective modules correspond to vector bundles as you already pointed out.

 - Let $R$ be the coordinate ring of a variety and $I$ a radical. Then the $R$ module $R/I$ corresponds to attaching a one dimensional vector space on each point of $Z(I)$ and the zero vector space everywhere else. For Example $R=k[x,y]$ and $I=(x,y)$ gives the skyscraper sheaf at the origin. $I=(x)$ gives the trivial one dimensional bundle on the x-axis etc. If your Ideal is not a radical, the situation is slightly more complicated. $R/I$ can be thought of as the trival bundle on an infinitesimal neighborhood of Z(I). 

 - Finally any finitely generated module (more generally a coherent sheaves on variety) is build up of vector bundles on subspaces in the following way: There exists a stratification of spec(R) such that the module pulled back to the strata is a vector bundle.