Need help understanding Laplacian

[Richard Baker]

Wikipedia article for laplacian says that it is how much the average value of the function deviates from the value of the function but this doesn't makes sense to me.  Would it be the average value surrounding the point minus the value at the point divided by the area of the surface or the volume confined within?  A while ago I realized that the laplacian equals the curl of the contour line of the function, but I forgot my logic.  I know that the laplacian is the sum of second-order partial derivatives, but I would like to know what it means geometrically. 

Thank you.

[swansont]

Second derivatives give information about the curvature of a function, much like the first derivative tells you the slope.

[Richard Baker]

" Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(p)." Please explain this sentence from the Wikipedia entry for laplacian.  https://en.wikipedia.org/wiki/Laplace_operator