The curvature in general is measured as 1/r, where r is the radius of an inscribed circle.  Since mean curvature is the average of the two principal curvatures it has the units of 1/mm.  The [[GaussianCurvature|Gaussian curvature]] is the product of them, so it is 1/mm^2.

see [[http://mathworld.wolfram.com/MeanCurvature.html|Mean Curvature]]

also [[http://brainvis.wustl.edu/wiki/index.php/Folding/Measurements|Van Essen Lab Folding/Measurements]]