The most popular stereograms are hidden images (SIRDS), repeating image patterns and stereo images. These are also called autostereograms as no special equipment like 3D glasses or stereoscopes are needed.

With hidden images the viewer tries to see a hidden object by staring at a picture of some kind repeating pattern. To create hidden images you need a software to modify the pattern with a depth map. A depth map is a grayscale image of the object you try to find in the final image. This object is usually a three dimensional model but can also be a "hand made" image. It is a lot harder to create depth map images by hand though.

Stereo images are usually photos or 3D computer renderings. The idea is simply to create two slightly different images of the same view - one image for the left eye to see and one fore the right eye. If it is done by camera you slightly shift the camera sideways between the takes. If it is done by computer rendering you shift "camera" in your scene in the 3D application. Then you place the two images side by side and the viewer tries to get the eyes to focus on the respective images (oh yes, I can feel a headache coming). One trick i have found useful is to put two empty toilet paper rolls together and use them as binoculars when looking at the stereo images.
If you want just one image we have the classic technique to use 3D glasses where one lens is blue and the other red. Then the respective images are also tinted in blue and red and instead of being placed side by side the images are merged into one single image. The looking at the final image without the special glasses it will feel like seeing double (which you are doing, kind of).

My personal favorite stereograms are the ones with repeating images, as you can make them yourself with any graphic software you like (in theory you can even draw or paint them by hand if you like) and you do not need special equipment like 3D glasses to view the them.
By placing copies of an object on the same level you can trick the eyes to see the image in 3D. To enhance the effect you can adjust the spacing between the copies and it will appear as some object are further away from the viewer than others. To improve the effect even further you can add a couple of more lines of objects: Object closer to each other should be placed in front of the rows were objects are furher apart.