What are those magical picture toys...?

Question

... that change pictures as you shift your eyes from one point to another? And how do they work?

Answer 1

I assume you are talking about those plastic pictures that have a ribbed surface in front of them.

The two pictures are divided into strips that are then interleaved and printed as one sheet.  The ribbed plastic acts as a series of prisms.  Depending on your angle of view, you see either the light from one set of strips or light from the other.  Usually, you can see a mid-way position where you can see both, too.

Prisms work (in this context) by a phenomenon called Total Internal Reflection.  Light will bounce off the inner surface instead of passing through it, if the angel is right.  The picture you CAN'T see is getting internally reflected and being passed out at a different angle to the one you can see.

Answer 2

Fake holograms. They have multiple pictures interlaced with each other in vertical stripes. There is a stripe of picture A, then a stripe of picture B, then a stripe of picture C; ABCABCABCABC.... In front of that, you have a clear plastic layer with vertical prisms or cylinders pressed into it, like a Fresnel lens. Depending on the viewing angle, you see all the A's, or all the B's or all the C's.

The same effect can be created with a diffraction grating, which blocks you view in certain directions.