The Laplacian of a function $u$ at a point $x$ measures the average extent to which the value of $u$ at $x$ deviates from the value of $u$ at nearby points to $x$ (cf. the mean value theorem for harmonic functions). As such, it naturally occurs in any system in which some quantity at a point is influenced by the value of the same quantity at nearby points. (This also explains the link between Laplacians the Laplacian and Brownian motion (or random walks), in which one repeatedly travels from a point $x$ to a randomly selected nearby point to $x$.)
The notion of "nearby points" requires only that one have a Riemannian metric structure on the underlying space, and so the Laplacian is a natural Riemannian invariant (and also a conformal invariant in two dimensions). This makes it a useful operator-theoretic proxy for Riemannian or conformal structure, in particular allowing one to use spectral theory to start controlling the Riemannian or conformal geometry of a domain. Being invariant, it also has a good chance of commuting (or nearly commuting) with other interesting invariant operators (e.g. partial derivatives, Hecke operators, heat or wave operators, etc.), and so the spectral theory of the Laplacian is often useful in illuminating the spectral theory and dynamics of many other invariant objects.
