Ok, I can give some good insight into this question and its history, actually:
1) To resolve 3D images, currently our brain "fuses" two images together and uses them to interpret a distance. Notice that if you close one eye, and open the other, and then reverse them, an image in the distance will change only slightly, whereas an image very close to your eyes will change significantly (try it with your finger).
2) Knowing this, those "Magic Eye Pictures" are actually called "Random Dot Stereograms". What they do is take a 3D picture (usually in the form of a black & white gradient) and then change ever so slightly the pixels in the picture, moving some closer to each other, and some farther away from each other, in order to fool your eye into fusing a 3D image.
3) Therefore, the best way to view one is to fool your eyes into fusing the image, put two fingers at the image, usually about ... 1" apart, and cross your eyes until the fingers overlap, now you're about half way there. You have to find the same distance that the "spaced pixels" are spread apart, as I mentioned, usually about 1" to generate a good 3D-image.
4) As someone mentioned holograms, I thought I would touch on that. Simply put, a 2D image is a reproduction of the intensity of an image, whereas a hologram actually reproduces the entire three-dimensional wavefront -- it is as if they took the same light that the hologram reflected, transported it, and then re-emitted it to your eye.
I hope that helps.
-Ankur