Why is it important to study the eigenvalues of the Laplacian? Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve?
In particular, if $0$ is in the spectrum, does this tell us anything about the solvability of the Dirichlet or Poisson problem for the region? This second question is motivated by the following theorem that I found while flipping through Davies' Spectral Theory and Differential Operators. From the emphasis he puts on it and the amount of work he did to prove it, I feel it must be important, but I can't say why. 

Let $\Omega\subset \mathbb R^2$ be regular, and let $H$ be the
  Friedrichs extension of $\small -\triangle$ initially defined on
  $C^\infty_c(\Omega)$. Then $0\in \operatorname{Spec} H$ if and only if
  the inradius of $\Omega$ is infinite.

 A: The spectrum of $-\Delta$ reveals properties of light, heat, sound, and atomic phenomena. I'll briefly summarize.
The heat equation
$$
           \frac{\partial u}{\partial t} = c\Delta u,
$$
can be solved if you have a complete basis of eigenfunctions for $\Delta u$. The eigenvalues give the time decay rates of the eigenfunctions in time. More explicitly, if $(-\Delta) e_{\lambda} = \lambda e_{\lambda}$, then
$$
          u(x,t)=e^{-c\lambda t}e_{\lambda}(x)
$$
is a solution of the heat equation, and, in principle, a full solution can be built from linear (discrete or continuous) sums of such solutions. This comes from the classical problem of separation of variables originally proposed by Fourier in his Treatise on Heat Conduction. Spectral Theory has its origins here.
In a similar way, one may develop solutions of the wave equation:
$$
            \frac{\partial^{2} u}{\partial t^{2}}=\frac{1}{c^{2}}\Delta u.
$$
The wave equation governs the time-dependent behavior of the electric field in a homogenous, isotropic, source-free material. (The propogation of light is described through the wave equation.) In this case the eigenvalues determine the modulation frequency for a particular eigenfunction through separation of variables. This corresponds to classical notions of prism and spectrum. For a string fixed at two endpoints and undergoing small vibrations, the classical one-dimensional wave equation describes mechanical displacements, and gives a fundamental vibrational mode and all integer multiples of those modes (a.k.a. harmonics.) This relates to sound, and it's almost certain that this is the origin of the term "harmonic anaysis." The harmonic frequencies are derived from the eigenvalues of $\frac{d^{2}}{dx^{2}}$. A circular drum head is similarly analyzed from the eigenvalues and eigenvectors of the 2-d Laplacian. The resonant frequencies of the drum are non-harmonic and are derived from the eigenvalues of the Laplacian.
The non-relativistic hydrogen atom has a corresponding Quantum Mechanical Schrodinger Wave equation
$$
   i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2\mu}\Delta \psi + \frac{e^{2}}{4\pi \epsilon_{0}r}\psi .
$$
Though the problem is not fully reduced to eigenvalues/eigenvectors of $\Delta$, the eigenvalues of $\Delta$ on functions defined on the unit spherical shell (spherical harmonics) have to do with the permissible values of the orbital energies of the atom; this reveals much about the orbital shells, and the spectral lines of the Hydrogen atom. Remember S, P, D, F from Chemistry? http://en.wikipedia.org/wiki/Atomic_orbital.
There are applications to fluids as well.
