Since you ask about mean curvature, we should assume that you mean a surface that is embedded or at least immersed in Euclidean 3-space. Then my favorite definition of Gauss curvature at a given point p is the following:
Position the surface so that p is at the origin and the tangent plane is the xy-plane. By rotating the surface about the z-axis, you can always position it so that the surface is given by
z = a-(a x^2 + b y^2)/2 + higher order terms
Then a, b are known as the principal curvatures at p, relative to the unit normal given by (0,0,1) and K = ab is the Guass curvature. The mean curvature is given by H = a + b.
However, there are (at least) two possible ways to position the surface this way. You can also flip the surface upside down. This has the effect of flipping the signs of a and b. This flips the sign of H but not K.
Therefore, the Gauss curvature K is defined independent of the choice of the unit normal. It therefore is well defined on a non-orientable surface.
On the other hand, defining H on the whole surface requires a choice of the unit normal on the entire surface (or, equivalently, a choice of orientation on the surface itself). You can do this, but neither canonically nor continuously. So, I know only how to define mean curvature up to sign.
(EDIT by Deane: Missing minus sign and factor of 1/2 inserted into formula above. I never keep close track of these things. You can always work out details like that by checking a canonical example. In this case, it would be the unit sphere shifted down:
z = sqrt{1 - x^2 - y^2) - 1 = -(x^2 + y^2)/2 + higher order terms)