There are several definitions of the term curvature in mathematics, but I think the simplest explanation of the first point is in regards to the second derivative of a function. The Laplacian is really just the second derivative of the electric potential equal to zero. Think of a straight line. Such an object has a finite first derivative but zero second derivative. In contrast, a parabola will have finite first and second derivatives, and thus a finite curvature.
I think this is consistent with @Qmechanic's post, but he/she is much more adept at math than I so I will let them clarify.
Side Note: The Poisson equation for an electrostatic potential, for comparison, can have curvature under the definition I used.