Eigenvectors, i.e., principal components, are vectors that are un-correlated and that contain loadings - what they summarise in your data is covariance among your variables. So, when you include eigenvectors as covariates in regression models, you are adjusting your regression coefficients based on these. It has been found that doing this can help to control for population structure in genetic studies. You should decide on the number of eigenvectors (PCs) to include as covariates via manual inspection of pairwise bi-plots, as per:

• A: PCA plot from read count matrix from RNA-Seq
• Produce PCA bi-plot for 1000 Genomes Phase III in VCF format (old)

For example, in the following figure, eigenvectors 1, 2, an 3 (PCs 1, 2, and 3) are clearly segregating the different populations; so, we would include these eigenvectors as covariates in modelling if we were studying some disease trait / phenotype that spanned all populations

Kevin