Stereogram
Stereogram is a pair of two-dimensional panels depicting the view of a scene or an object from the vantage points of the right and left eyes. Observing the panels superimposed in a stereoscope results in the experience of three-dimensionality by virtue of the fact that object depth is encoded as right/left position difference in the panels. Because in a stereoscope one views a representation rather than a real scene, the word illusion is sometimes associated with a stereogram.
Originally, stereogram referred to a pair of stereo images which could be viewed using a stereoscope. Other types of stereograms include anaglyphs and autostereograms. The stereogram was discovered by Charles Wheatstone in 1838. He found an explanation of binocular vision which led him to construct a stereoscope based on a combination of prisms and mirrors to allow a person to see 3D images from two 2D pictures. Oliver Wendell Holmes, Sr. invented an improved form of stereoscope in 1861, which had no mirrors and was inexpensive to produce. These stereoscopes were immensely popular for decades.
Stereograms were re-popularized by the creation of autostereograms on computers, wherein a 3D image is hidden in a single 2D image, until the viewer focuses the eyes correctly. The Magic Eye series is a popular example of this. Magic Eye books refer to autostereograms as stereograms, leading most people to believe that the word stereogram is synonymous with autostereogram. Salvador Dalí created some impressive stereograms in his exploration in a variety of optical illusions.
An auto stereogram is an optical illusion of depth usually observed by allowing the eyes to focus behind the image (diverge), or, less often, in front of it (converge). These two methods are also known as wall-eyed and cross-eyed, respectively. The slight differences in vertical repetitions of figures or random dots create the illusion of depth in the 2D image, just as the slight difference in perspective between one's eyes creates the perception of depth on 3D objects and scenes.
According to Magic Eye, a maker of autostereograms, "most people prefer the diverging method". However, with normal stereograms, this imposes a limit on the size of the image, since there is a limit to how much the eyes diverge; images created for the cross-eyed method can be larger. If a stereogram is viewed with the wrong method, the depth information is seen 'reversed'; points intended to be in the background appear in the foreground and vice versa.

Practical uses
While stereograms have typically been used for amusement, including "3D" movies using anaglyph motion pictures, posters and books of autostereograms, and historical replicas of early stereograms, there are also practical uses for technology.
The Mars Exploration Rovers, launched by NASA in 2003 to explore the surface of Mars, are equipped with unique cameras that allow researchers to view stereoscopic images of the surface of Mars.
The two cameras that make up each rover's Pancam are situated 1.5m above the ground surface, and are separated by 30 cm, with 1 degree of toe-in. This allows the image pairs to be made into scientifically useful stereoscopic images, which can be viewed as stereograms, anaglyphs, or processed into 3D computer images.
Curious rock with a jutting portion at "Home Plate" via the Mars Spirit Rover.
The ability to create realistic 3D images from a pair of cameras at roughly human-height gives researchers increased insight as to the nature of the landscapes being viewed. In environments without hazy atmospheres or familiar landmarks, humans rely on stereoscopic clues to judge distance. Single camera viewpoints are therefore more difficult to interpret. Multiple camera stereoscopic systems like the Pancam address this problem with unmanned space exploration.
In a clinical use Stereograms cards are frequently used
By Orthoptists and vision therapists in the treatment of many binocular vision and accommodative disorders.
In a Mathematical, scientific and engineering use the stereo pair image of the lake, stereopair photographs are sometimes used to help visualize aerial photographs. Cartographers may also generate stereopairs using Computer programs in order to visualize topography in three dimensions. In biology and chemistry, complex molecular structures are often rendered in stereopairs. The same technique can also be applied to any mathematical (or scientific, or engineering) parameter that is a function of two variables, although in these cases it is more common for a three-dimensional effect to be created using a 'distorted' mesh or shading (as if from a distant light source).

A Look into the History of the Stereogram

One eye can only give a 2-dimensional image. Stereopsis, the vision of the third dimension (depth), is obtained through several automatic processes; one of these consists in comparing the differences of the images that each eye produces. Let's take this example:

If you look at this image through 3D-glasses (red filter for the left eye, blue filter for the right eye), then each eye sees a square, the square of the right eye being slightly shifted to the left. With both eyes together, you will have the impression to see a floating square (coming out of the screen)!

This effect was demonstrated for the first time in 1838 by Wheatstone.

There is a more sophisticated variant, where the red and blue filters of the glasses are replaced by horizontal and vertical polarizing filters, and the two polarized images are projected on a cinema screen.
In 1959
Dr. Bela Julesz studied stereopsis with images like this:

In the last picture Start with a surface filled with random dots, cut out a small square, move it a bit to the left, fill the resulting white space on the right with new dots.
The image pair is viewed with a stereoscope so that the left eye sees the original image, and the right eye sees the transformed image: both images fuse to one single image where a square seems to float above the dotted surface.
(You may even arrive in seeing the floating square without a stereoscope, if you look with each eye at one image, and squint so that both images fuse to one; but this exercise is very straining for the eyes!)
The reason to use random dots was to prove that the brain can produce a 3-dimensional image, even when the original image doesn't contain any suggestive forms.

In 1979
Christopher Tyler found a trick which made it possible to use only one image for both eyes, like in this example:

Instead of a dotted surface, we start with a background image consisting of a vertical stripe large of about 4 cm.
This stripe is copied several times from right to left, but according to the object to represent, the copied points are shifted more or less.
If you look relaxed at this image, and try to focus on a fictive point behind (or in front of) the image, then there will (hopefully) come a moment when your eyes get confused by the repeating stripes and will see only 7 of them instead of 8. This is somehow the contrary effect of squinting.

A BRIEF HISTORY OF THE INVENTOR OF THE STEREOGRAM SIR CHARLES WHEATSTONE
Sir Charles Wheatstone Was born 6 February 1802 to 19 October 1875,and was an English scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope (a device for displaying three-dimensional images), and the Playfair cipher(an encryption technique). However, Wheatstone is best known for his contributions in the development of the Wheatstone bridge, originally invented by Samuel, which is used to measure an unknown electrical resistance, and as a major figure in the development of telegraphy.
Charles Wheatstone was born in Barnwood Gloucester. His father was a music-seller in the town, who moved to 128 Pall Mall, London, four years later, becoming a teacher of the flute.
Charles, the second son, went to a village school, near Gloucester, and afterwards to several institutions in London. When he was about fourteen years old he was apprenticed to his uncle and namesake, a maker and seller of musical instruments such as the Wheatstone concertina, but he showed little taste for handicraft or business, and loved better to study books.
His father encouraged him in this, and finally took him out of the uncle's charge.

Wheatstone was knighted in 1868, after his completion of the automatic telegraph. He had previously been made a Chevalier of the Legion of Honor. Some thirty-four distinctions and diplomas of home or foreign societies bore witness to his scientific reputation. Since 1836 he had been a Fellow of the Royal Society, and in 1859 he was elected a foreign member of the Royal Swedish Academy of Sciences, and in 1873 a Foreign Associate of the French. The same year he was awarded the Ampere Medal by the French Society for the Encouragement of National Industry. In 1875 he was created an honorary member of the Institution of Civil Engineers. He was a D.C.L. of Oxford and an LL.D. of Cambridge.
Stereopsis was first described by Wheatstone in 1838. In 1840 he was awarded the Royal Medal of the Royal Society for his explanation of binocular vision, a research which led him to make stereoscopic drawings and construct the stereoscope. He showed that our impression of solidity is gained by the combination in the mind of two separate pictures of an object taken by both of our eyes from different points of view. Thus, in the stereoscope, an arrangement of lenses or mirrors, two photographs of the same object taken from different points are so combined as to make the object stand out with a solid aspect. Sir David Brewster improved the stereoscope by dispensing with the mirrors, and bringing it into its existing form with lenses.
The 'pseudoscope' (Wheatstone coined the term from the Greek ψευδίς σκοπειν) was introduced in 1852, and is in some sort the reverse of the stereoscope, since it causes a solid object to seem hollow, and a nearer one to be farther off; thus, a bust appears to be a mask, and a tree growing outside of a window looks as if it were growing inside the room. Its purpose was to test his theory of stereo vision and for investigations into what would now be called experimental psychology.

A Continued History

Leonardo da Vinci realized that objects at different distances from the eyes project images in the two eyes that differ in their horizontal positions, but had concluded only that this made it impossible for a painter to portray a realistic depiction of the depth in a scene from a single canvas. Leonardo chose for his near object a column with a circular cross section and for his far object a flat wall. Had he chosen any other near object, he might have discovered horizontal disparity of its features. His column was one of the few objects that projects identical images of itself in the two eyes.

Stereoscopy became popular during Victorian times with the invention of the prism stereoscope by David Brewster. This, combined with photography, meant that tens of thousands of stereograms were produced.
Until about the 1960s, research into stereopsis was dedicated to exploring its limits and its relationship to singleness of vision. Researchers included Peter Ludvig Panum, Ewald Hering, Adelbert Ames Jr., and Kenneth N. Ogle.
In the 1960s, Bela Julesz invented random-dot stereograms. Unlike previous stereograms, in which each half image showed recognizable objects, each half image of the first random-dot stereograms showed a square matrix of about 10,000 small dots, with each dot having a 50% probability of being black or white. No recognizable objects could be seen in either half image. The two half images of a random-dot stereogram were essentially identical, except that one had a square area of dots shifted horizontally by one or two dot diameters, giving horizontal disparity. The gap left by the shifting was filled in with new random dots, hiding the shifted square. Nevertheless, when the two half images were viewed one to each eye, the square area was almost immediately visible by being closer or farther than the background. Julesz whimsically called the square a Cyclopean image after the mythical Cyclops who had only one eye. This was because it was as though we have a cyclopean eye inside our brains that can see cyclopean stimuli hidden to each of our actual eyes. Random-dot stereograms highlighted a problem for stereopsis, the correspondence problem. This is that any dot in one half image can realistically be paired with many same-coloured dots in the other half image. Our visual systems clearly solve the correspondence problem, in that we see the intended depth instead of a fog of false matches. Research began to understand how.
In the 1970s, Christopher Tyler invented auto stereograms, random-dot stereograms that can be viewed without a stereoscope. This led to the popular Magic Eye pictures. In 1989 Medina demonstrated with photographs that retinal images with no parallax disparity but with different shadows are fused stereoscopically, imparting depth perception to the imaged scene. He named the phenomenon "shadow stereopsis." Shadows are therefore an important, stereoscopic cue for depth perception. He showed how effective the phenomenon is by taking two photographs of the Moon at different times, and therefore with different shadows, making the Moon to appear in 3D stereoscopically, despite the absence of any other stereoscopic cue.

Geometrical Basis for Stereopsis

Stereopsis appears to be processed in the visual cortex in cells having receptive fields in different horizontal positions in the two eyes. Such a cell is active only when its preferred stimulus is in the correct position in the left eye and in the correct position in the right eye, making it a disparity detector.
When a person stares at an object, the two eyes converge so that the object appears at the center of the retina in both eyes. Other objects around the main object appear shifted in relation to the main object. In the following example, whereas the main object (dolphin) remains in the center of the two images in the two eyes, the cube is shifted to the right in the left eye's image and is shifted to the left when in the right eye's image.

Because each eye is in a different horizontal position, each has a slightly different perspective on a scene yielding different retinal images. Normally two images are not observed, but rather a single view of the scene, a phenomenon known as singleness of vision. Nevertheless, stereopsis is possible with double vision. This form of stereopsis was called qualitative stereopsis by Kenneth Ogle.
If the images are very different (such as by going cross-eyed, or by presenting different images in a stereoscope) then one image at a time may be seen, a phenomenon known as binocular rivalry.
Not everyone has the same ability to see using stereopsis. One study shows that 97.3% are able to distinguish depth at horizontal disparities of 2.3 minutes of arc or smaller, and at least 80% could distinguish depth at horizontal differences of 30 seconds of arc.