Here is a brief description of several applications, which can be divided into elementary and advanced:
Geometric: An n x n matrix can be thought of as a geometric transformation in R^n, for example any 2 x 2 matrix is characterized by how it transforms the points on a unit circle. In practice the properties of matrices make are such that the circle will be variously stretched or shrunken (even collapsed) along each of two directions. The directions along which pure stretch or shrink occur are the eigenvectors of the matrix, and the eigenvalues are the scaling factors in those dimensions.
Rotational Dynamics: A general 3 dimensional object without any rotational symmetries will have its moment of inertia be a 3x3 matrix which happens to be symmetric i.e. M = Transpose(M). The eigenvectors of this matrix are called "principle directions" and these are directions passing through the center of mass along which the object can rotate. Each eigenvalues is the moments of inertia for that axis.
Linear Oscillations: Any system of coupled linear oscillators (interconnected springs in n dimensions) can be represented as a matrix for which the components of the eigenvectors are modal amplitudes for each oscillator and the eigenvalues are modal frequencies. This is a corrolary to the next application:
Linear Differential Equations with Constant Coefficients can always be solved exactly with eigenvectors and eigenvalues, and these can be used to represent chemical reactions, oscillations, beam bending, and many more.
Advanced:
The eigenvalue problems of infinite dimensional self-adjoint matrices correspond to 2nd order Linear differential equations with boundary conditions satisfying certain properties. The solutions are functions which correspond to infinite dimensional vectors and most importantly the solution set forms a basis for a function space typically inclusive of the relevant initial conditions. An example is Fourier series of sines and cosines which has been applied to wave motion and diffusion and many more.
The standard formulation of quantum mechanics is based on an infinite-dimensional space of vectors and operators. Every physical property corresponds to an operator, and observable values are the eigenvalues and the eigenfunctions are pure states for that property.
The use of eigenvectors really took off with quantum mechanics, but since then there have been enough published applications to fill an entire library shelf section with back volumes of a thick green journal that I have not read all of yet.