I think of the Laplacian as the rate at which a function at a point differs from the average at the points surrounding it. In fact, in n dimensions, it can be written
\Delta f (x) = 2n \lim_{h\to 0} \frac{\operatorname{Avg}\{f(z) :\; |z-x|=h\}-f(x)}{h^2}
(I may be off by a constant factor)...