From what I understand, just \nabla(f) describes the gradient of a function f -- meaning all the first order partial derivatives (of a certain point) in a vector.

\nabla * f describes the divergence of the function f -- meaning a scalar field of the quantity of the vector field's sources at each point.

\nabla * \nabla(f), divergence of gradient, then describes the Laplace operator. Also written as \nabla^2(f) or \Delta(f) -- note that it's an uppercase delta.