The role eigenvalues play in a factor analysis is similar to the role they play in principal components analysis: they allow you to know how much variation each factor or component can explain. The goal in selecting eigenvalues is to include enough variation in your model that the end solution is valid, while not overcomplicating your model with too many factors.
The scree plot is a graphical approach to selecting eigenvalues. This approach, which was developed by Raymond Cattell in 1966, is a somewhat subjective means of selecting factors. The scree plot places the eigenvalues on the y-axis and the factors on the x-axis. The user of this procedure will find an "elbow" in the scree plot, which is a point after which all the eigenvalues are aligned in a linear fashion. The eigenvalues before this elbow are those that the researcher should use in the factor analysis.
Kaiser's Rule is an objective decision-making rule for the selection of eigenvalues. It states that all eigenvalues exceeding unity should be used. This rule is intuitively satisfying in that any factors associated with eigenvalues under unity contain less information than the original variables that were used in the factor analysis. This rule is both objective and easy to use.
Horn's Procedure recognizes that the processes of factor analysis and principal components analysis will exploit the random variation in the data. This capitalization on random variation leads to the first eigenvalue always being greater than unity, regardless of whether the data itself has any interesting correlations among variables. Horn's Procedure addresses this problem by comparing eigenvalues not to unity, but to the eigenvalues of a principal components analysis that use purely random, uncorrelated data. Each eigenvalue from the factor analysis is compared to the corresponding eigenvalue for the principal components analysis. If the factor analysis eigenvalue is larger, it is chosen. Otherwise, it is discarded.