In 1947, the first instant camera was built. What was so special about it was the fact that the film developed itself in only a few minutes. This was a major achievement considering that it used to be fairly complicated and time consuming to develop film. Even though it seems to be magic that makes the pictures appear, it actually is a fairly simple process.
There are not too many differences between a regular camera film and an instant camera film, thus, I will commence with explaining photography in general.
Usually, film consists of a plastic base coated with a silver compound. When light hits the film it causes a chemical reaction that forms silver atoms. The pattern created by the silver compound that has reacted with very much, very little or no light at all shows the image taken. Colour film is very similar, the only difference is that it has three layers, one sensitive to blue light, one sensitive to green light and one sensitive to red light. When the film is exposed to light, each layer reacts with its specific colour. Again, the pattern created by the reaction shows the picture taken.
While the chemicals needed to develop normal film have to be added one by one, the instant camera has all the required chemicals in the film itself.

Underneath each colour layer, there is a developer layer containing the dye chemicals for the specific colour. These light sensitive and developer layers lie in between a black base layer and the image layer, the timing layer and the acid layer. The different layers are components of a chain reaction waiting to be set in motion. The component triggering the chain reaction is called reagent layer and is a mixture of opacifiers that block light, alkali that neutralizes acid, white pigment and some more elements. It is located in between the image layer and the light sensitive layer. When a picture is taken, the film is exposed to light for a short period of time and then pressed through a pair of rollers that releases the reagent layer through pressure. The reagent chemicals change the particles exposed to light into metallic silver and dissolve the dye that make the colours of the photo appear. At the same time the reagent chemicals also move upward and activate the layers on top. Once all processes are completed, the light blocking chemical clears up and the picture will slowly appear.
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http://science.howstuffworks.com/innovation/everyday-innovations/instant-film.htm