The Laplace operator is to Analysis and PDEs is (almost) like what a sum of squares is to Linear Algebra and Statistics.
The Laplacian is the simplest differential quadratic form corresponding via the Fourier transform to the square of the Euclidean distance. This may explain in part its fundamental role in the Harmonic Analysis on Euclidean spaces.
The Laplacian (or, more precisely, $\frac{1}{2}\Delta$) is the infinitesimal generator of a Brownian motion on $\mathbb R^n$, which is the simplest and most ubiquitous of the continuous-time stochastic processes.
The Laplace-Beltrami operator on a Riemannian manifold is conformal invariant. The connection between harmonic functions, complex analysis and probability is the most tight in dimension 2 (Levy's theorem, Schramm-Loewner evolution, ...).
The Laplace operator is the trace of the coefficient of the quadratic term in a local Taylor expansion of a function (the Hessian matrix). This implies that it will pop up (together with the determinant of the Hessian matrix) in many problems related to optimization.
