Understanding the Laplace operator conceptually The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it?
Any good essays (combining both history and conceptual understanding) on the Laplace operator, and its subsequent variations (e.g. Laplace-Bertrami) that you would highly recommend?

Here is some intuition:
I think the most basic thing to know about the Laplacian $\Delta$ is that $-\Delta = -\text{div} \, \nabla$, and $-\text{div}$ is the adjoint of $\nabla$.  Hence, $-\Delta$ has the familiar form $A^T A$ which recurs throughout linear algebra.  We see that $-\Delta$ is a self-adjoint positive semidefinite operator, and so we would expect (or hope) that the familiar properties of positive semidefinite operators in linear algebra hold true for $-\Delta$.  Namely, we expect that $-\Delta$ has real nonnegative eigenvalues, and that there should exist (in some sense) an orthonormal basis of eigenfunctions for $-\Delta$.  This provides some intuition or motivation for the topic of "eigenfunctions of the Laplacian".  (By the way, I think the Laplacian should have been defined to be $- \text{div} \, \nabla$.)
Notice that the integration by parts formula can be interpreted as telling us that $-\frac{d}{dx}$ is the adjoint of $\frac{d}{dx}$ (in a setting where boundary terms vanish).  Fourier series can be discovered by computing the eigenfunctions of the anti-self-adjoint operator $\frac{d}{dx}$ in an appropriate setting.  Moreover, a multivariable integration by parts formula can be interpreted as telling us that $-\text{div}$ is the adjoint of $\nabla$.  Green's second identity can be interpreted as expressing the self-adjointness of the Laplacian.
