Things that vibrate have natural modes of vibration. A particular vibrational pattern can be decomposed into a time-varying linear combination of these modes. The modes of vibration are eigenfunctions and the frequencies at which they vibrate are the square root of the corresponding eigenvalues.

You can look up a vibrating drum head (circular membrane) for a simple example.

Landau/Lifshitz: "Mechanics" has a chapter on small oscillations

That's more of a further packing than an unpacking. Although that totally should be an expression for things that go on for too long: "Can you pack this for me please"

Theory of Vibration with Applications by William Thompson and Marie Dillon Dahleh.

Say you have two cars linked, with some spring constant;

| --^^-- [c1] --^^-- [c2] --^^--|

where '^^' is a spring and '|' is a wall.

The motion of these cars can be written using the spring forces in the system or, alternately, as the harmonic motion of the undamped system with some natural frequency.

Setting this up as two simultaneous equations (one for each car) and solving for the roots give you the eigenvalues. The natural frequency is the square root of the eigenvalue. In other words, the eigenvalues help you define the natural frequencies which can be used to characterize the motion of the more complicated spring-mass system.