For questions involving calculation or interpretation of eigenvalues or eigenvectors.
Questions involving the calculation or interpretation of eigenvalues should use eigenvalues. This may include factor analysis, principal components analysis or regression, or other model estimation functions that require a positive definite matrix (of which all eigenvalues are positive). In factor analysis, a factor's eigenvalue $=\sum($loadings on that factor$)^2$; eigenvalue $\div\sum$ eigenvalues $=$ % total variance explained by the factor.
An eigenvector of a square matrix A is a non-zero vector v that, when the matrix is multiplied by v, yields a constant multiple of v, the multiplier being commonly denoted by λ. That is:
A v = λ v
The number λ is called the eigenvalue of A corresponding to v.
In the context of factor analysis, a factor's eigenvalue is the sum of all variables' squared loadings on that factor. The factor loading is the correlation of the variable with the factor. The squared loading is the variance explained in the variable by the factor. The factor's eigenvalue divided by the sum of all eigenvalues is the proportion of total variance explained by the factor. From Wikipedia:
If a factor has a low eigenvalue, then it contributes little to the explanation of variances in the variables and may be ignored as redundant with more important factors.
All of the above is also true of principal components analysis. Principal components regression eliminates components with small eigenvalues for the similar purpose of reducing the dimensionality of a set of regressors.
Many criteria for identifying an appropriate threshold exist, and vary in their utility.