An eigenvalue corresponding to an eigenvector satisfies the following relationship.

To see this from the equation you supplied, just multiply it out and move the negative part to the other side and you will get what I wrote above.

If you are given the eigenvector v you just multiply it by A (Av), see what you get and if it is an eigenvector, there will be some scalar multiple lambda that you can multiply your vector by to get the same thing, as in the above equation.

The equation in the form you supplied is key for actually solving what these eigenvalues and vectors are. You will subtract lambda from the diagonal and take the determinant. Set this equal to 0 and this is your characteristic polynomial. The roots of this polynomial will be your eigenvalues. Then to find the corresponding eigenvectors you simply subtract those roots from the diagonal and solve the equation for v.