eigenvalue
A number and matching direction that a matrix only stretches, never rotates.
For a square matrix A, an eigenvector \mathbf{v} is a nonzero vector whose direction A leaves unchanged, only rescaling it by a number: A\mathbf{v} = \lambda\mathbf{v}. That number \lambda is the matching eigenvalue. For the symmetric covariance matrices PCA builds, the eigenvalues are always real, and the eigenvalue's size measures exactly how much of the data's spread lies along its eigenvector's direction -- which is why sorting eigenvalues from largest to smallest ranks the eigenimages (Section 6) from most to least informative.
Examples
If A\mathbf{v} = 3\mathbf{v}, then \mathbf{v} is an eigenvector of A with eigenvalue 3: A stretches \mathbf{v} to three times its length without changing where it points.