http://www.ncatlab.org/nlab/show/flat+functor
See http://ncatlab.org/nlab/show/flat+morphism for notions of flatness and link to flatness in derived algebraic geometry.
http://mathoverflow.net/questions/51095/flat-module-and-torsion-free-module
In the intro to the Soule et al Arakelov book, there is an explicit description of what flatness means for an arithmetic scheme.
nLab page on Flat