What is a fiber bundle? (for non-mathematicians)

How can I explain the concept of a fiber bundle to someone with no mathematical background?

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Draw a picture! For example you could use:

• the cylinder for a (trivial) $S^1$-bundle over $\mathbb{R}$
• the Möbius strip for a (non-trivial) $(0,1)$-bundle over $S^1$
• the volume between two spheres of different radius for a (trivial) $(0,1)$-bundle over $S^2$
• the torus for a (trivial) $S^1$-bundle over $S^1$

I think that those can already give some kind of intuition on what a fibre bundle is, in particular the Möbius strip is an easy non-trivial example. You can explain how it is different from the others by noticing that you cannot deform the Möbius strip into a cyilinder.

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Also, it would be helpful to talk about a Cartesian product. Some examples might be best made physical -- like all the configurations of an hour hand and a minute hand on a clock as a "physical" $S^1 \times S^1$. –  Ryan Budney Aug 13 '13 at 17:27