As explained by SandeepJ, "eigen..." is related to the spectrum of something. In particular, when one calculates the eigenvalues and their corresponding eigenvectors from A.x = lambda*x, adding the same vector x times a constant to both sides of the equation (say tau, i.e., one adds tau*x), only the eigenvalues get shifted by an amount equivalent to tau, but not the eigenvectors. The corollary that one can get from this statement, is that "eigen..." is about locating some characteristic functions or operators, which get unchanged under the shift of a constant, or something similar. We can state it in another way: in the process of studying "eigen..." we want to find some functions on the spectrum of some sort of functional, which remain invariant under certain types of transformation. This last phrase may sound pretty vague, but the thing is that "eigen..." is not something for which we can always define a straightforward algorithm.
The place which I liked the most for answering this question of yours is "Numerical Recipes in Fortran. The Art of Scientific Computing" (William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery)