[FabHK]

My intuitive take:

Eigenvectors answer the question, if you have a linear mapping from a space to itself: which lines through (through the origin) remain unchanged after the mapping? And the associated eigenvalues tell you how much a specific point on that line moves along the line. For example: If you rotate things in 3D, the rotation axis remains unchanged. That's the eigenvector. If you reflect things through a mirror plane, any line in that mirror plane will remain unchanged (with eigenvalue 1). Other lines will be, well, mirrored, but there is one line that is reflected onto itself: the one perpendicular to the mirror plane. However, points on it will go from one side to the other, so have eigenvalue -1.

The SVD says something else: Any linear mapping from a space (to itself or a different one - higher or lower dimensional) can be expressed as 1) first a rotation in the old space, then 2) some coordinate scaling or dilation (expansion/contraction) along the coordinate axes, described by the diagonal matrix with singular values, 3) then another rotation in the new space.