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

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.

I would say draw an example of a space which can be exhibited as a fiber bundle, draw in the base space as a subspace (if this is possible) and then show that for a sufficiently small neighbourhood around a point in the base space, the fiber of the neighbourhood just looks like the product of the neighbourhood and the fiber. Some easy starting examples would be a compact product space (why not pick a square, cylinder and a torus - all easy to draw) to show that there are trivial bundles, and then something a bit more exotic like a Möbius strip or a covering space (the connected double cover of a circle seems like a good start).

The point is that examples are definitely the best way to get the point across.