What are the importances of eigen values and eigen vectors?

The reason why eigenvalues are so important in mathematics are too many. Here is a very short and extremely incomplete list of the main applications I encountered in my path and that are coming now in mind to me:

Theoretical applications: 

• The eigenvalues of the Jacobian of a vector field at a given point determines the local geometry of the flow and the stability of that point; 
• An iterative method yk+1=Ayk is convergent if the spectral radius ρ(A) (the maximum absolute value of the eigenvalues of A) is < 1. 

Practical applications:

• Google Page Rank: The order in which your search results appear in Google is determined by computing an eigen vector. 
• Face Recognition: You can automatically recognize faces by computing eigenvectors of images. 
• In mechanics, the eigen vectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
• Geology and glaciology: In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. 
• Molecular orbitals: In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. 
• Vibration analysis: Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom.

And there are also other type of eigenvalue problems, more difficult to solve, e.g. Generalized and Nonlinear Eigenvalue Problems, with even more interesting applications.