An eigenvalue for a matrix A is a scalar λ such that:
Ax = λx
As a simple example, consider the matrix A:
[ 3 1 ]
[ 0 2 ]
Manipulate the equation above to get:
det(A-λI) = 0
Take the determinant
| 3-λ 1 |
| 0 2-λ |
Giving:
det(A-λI) =
(3-λ)(2-λ) = 0
Solving gives the eigenvalues 2 and 3.
Using the eigenvalues, now you can find the eigenvectors.
For an n by n matrix, there will always be n eigenvalues, but they may not be distinct. Additionally, the sum of all eigenvalues is equal to the trace of the matrix.