The Laplacian is the div of the grad of a function on Euclidean space:
$$ \nabla^2 \psi = \nabla \cdot \nabla \psi = \Delta \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} $$
The Laplacian represents the flux density of the gradient flow of a function, and can be thought of as a measure of source or sink in the flow.
Laplace's equation is the case when the Laplacian equals zero: $\nabla^2 \rho = 0$.
