Eigenvector
Definition
The special directions a transformation leaves pointing the same way — merely stretched or flipped.
Technical definition
A nonzero v satisfying Av = λv. Principal axes of inertia and normal modes of vibration are eigenvectors.
Example
A spinning top's stable spin axes are eigenvectors of its inertia tensor — spin it about another axis and it wobbles.
Common confusion
Eigenvectors aren't unique — any nonzero multiple counts. The direction matters, not the length.