Curvature measures how fast a curve is changing direction at a given point. So, local bending of the surface is measured by curvature. There are a few types of curvatures useful for you:

Normal curvature is the curvature of the curve projected onto the plane containing the curve’s tangent (T) and surface unit-normal (u) vectors. In other words, it is the curvature of the plane containing both u and T. All curves with the same tangent vector will have the same normal curvature.

Taking all possible tangent vectors then the max (k₁) and min (k₂) value (extremum) of the normal curvature at a point are called principal curvatures.

Mean curvature is equal to average of principal curvatures (k_H = (k₁ + k₂) / 2).

Gaussian curvature is product of principal curvatures (k_G = k₁ * k₂). Sign of the Gaussian curvature can be used to characterize the surface.

k₁ * k₂ > 0 tells how locally spherical or elliptic the surface is.

k₁ * k₂ = 0 tells how locally cylindrical or parabolic the surface is.

k₁ * k₂ < 0 tells how locally saddle-shaped or hyperbolic the surface is.

For more detail and discretization you can check Discrete Differential Geometry Operators for Triangulated 2-Manifolds by Meyer et al.