There are many different notions of dimension. I think Manin has a survey paper. Artin-Stafford use Gelfand-Kirillov dimension of an algebra in their Inventiones paper.
Dobbs LNM0147: Cech cohomological dimension for comm rings.
http://ncatlab.org/nlab/show/cohomological+dimension
http://ncatlab.org/nlab/show/homotopy+dimension
http://ncatlab.org/nlab/show/covering+dimension
de Jong on dimension of varieties
nLab page on Dimension