The art of producing the effect of stereoscopic vision, i.e. presenting a different image to each eye. This gives the illusion of depth perception, and a three-dimensional appearance.
There are several methods by which a different image can be sent to each eye. A stereoscope is the most direct solution to the problem - it is an instrument which simply allows one eye to see one picture, and the other eye to see a second one. The red plastic toy with the cardboard discs mentioned by jesler is a stereoscope. There are old-fashioned ones made of wood, with a visor to cover the eyes, lenses in front of each eye, a rod extending like Pinnochio's nose with a sliding stereograph holder and a divider so neither eye can see the picture intended for the other. To use one: find a brightly-lit place with no shadows nearby, put a stereograph in the holder (centred over the rod), look through the visor, and slide the holder until the pictures are in focus. The pictures will be magnified by the lenses to cover a larger field of view, so the experience is more immersive. Only one view of the 3D scene is possible, one cannot view a different angle. Larger stereoscopes are available, with mirrors and prisms for viewing bigger stereographs, like two successive aerial photographs for an overhead 3D view of some landscape.
Video stereoscopes are in use for virtual reality tools... very cool. With proper tracking and a fast enough processor, the images can be updated in real time according to the movements of the head, so one can view the scene from any angle.
Another way to give different images to each eye is to take advantage of refraction, in a method called integral photography. In the simpler type of integral photography, called lenticular arrays, a special picture is produced in stripes of about 1mm wide, then it is placed behind a washboard-shaped piece of transparent plastic (where the bumps on the washboard are also about 1mm wide). The line of sight from the left eye hits a bump in the washboard at a certain angle, and so one part of the stripe is visible, and the right eye hits the bump at a different angle, so gets refracted differently, so sees a different part of the stripe. One can view the same scene from different angles horizontally over a somewhat narrow range (looking up or down at the picture has no effect), and simple animation can be done in 3D by having the viewer move side to side. In the more sophisticated integral photography, instead of lines of hemicylinders there's a honeycomb of hemispheres. This allows the viewer to see a higher or lower angle on the scene, too.
Along similar lines, a mask consisting of a large number of narrow vertical black strips evenly spaced on a transparent plate can be placed in front of an image (like a diffraction grating, but the lines and gaps are wider), in a method known as parallax barrier. The left eye sees through a gap at a certain angle and sees part of the image, the right eye sees through the same gap at a different angle and so sees a different part of the image. This allows the viewer a little mobility with respect to the display, but not much. The mask can be put in front of a television or computer monitor to show video in 3-D.
One way to seperate one image into two is with colour. While this is a lossy method of combining images, it is quite effective thanks in part to the capacity of the brain to infer more sense than is actually present. High-quality inks and filters are required to make this method work well. Typically, the image for the left eye is printed only in shades of red, and the image for the right eye is printed in green and blue. A red filter is put over the left eye, and a green filter (which usually lets through some blue too) is put over the right eye. If the printing process is CMY-based, the cyan ink must look the same as white through the left filter, whereas the magenta and yellow inks must look the same as red (and black) through the left filter. Similarly, cyan should (ideally) look the same with or without the right filter, and magenta should look the same as blue and cyan should look the same as green through the right filter. The eye and brain make up the missing details by matching corresponding light and dark parts of each image, and matching corresponding edges. The brain gets more colour information from green than red, and more people have the right eye dominant over the left eye, so the green filter is normally put over the right eye. This is called an anaglyph. Video anaglyphs used to be shown in movie theatres, though the novelty has long since worn off. There is an occasional video anaglyph on television, and some static ones printed in magazines and comics from time to time. Only one angle onto the scene is possible.
Another way is to make a random dot stereogram. This method does not allow colour or shade at all, the only effect visible in a RDS is depth. A random pattern, which (for the most part) has a horizontal periodicity of about the same as an average person's interocular distance (distance between one's eyes), has specific flaws in the periodicity to give the impression of depth. Only one angle on the scene is viewable. A video RDS is highly unadvisable.
Another way is using polarisation. I don't know the precise mechanics of how this works, I think two projectors are used with a horizontal polarization filter over one and a vertical filter for the other. One image for each eye is projected onto a specially-prepared screen, which keeps the polarization of each image intact. The audience then wears glasses, with horizontal and vertical polarisation over opposite eyes, and the stereoscopic effect is achieved without losing any colour or resolution. Only one view of a scene is possible, but video is relatively simple once the synchronisation problem is solved.
Yet another way is to use time. A pair of special glasses, with the lenses for each eye electronically triggered to switch between transparent and opaque, is connected to a control unit on a display (usually a CRT or some other fast display, LCDs are very bad for this because of the shadow that remains when the display is changed suddenly). At the time the display shows the image for the left eye, the lens for the right eye is opaque and the lens for the left eye is transparent, and vice versa. The images are switched fast enough that persistence of vision fills in the blanks and a smooth 3D animation is observed. Again, only one angle on the scene is possible. Video or animation is almost as easy to accomplish with this method as static pictures.
There are many more cues for the eye to perceive depth, all listed under jprockwell's writeup under depth perception. By far the most common target for stereoscopy is binocular disparity, as this is the single most effective (and easiest) cue for depth perception. Yet more information on stereoscopy is available at the node autostereoscopic display.
Thanks to jprockwell for improvements and additions to this writeup.