Let's Get Technical
The film used in a Polaroid camera is similar to that used in a standard 35 mm camera. The difference lies within the film itself; there are certain Polaroid-specific elements included that allow the "magic" to happen. When film is developed, it is put through a specific process that involves the use of chemicals, developers, dyes and couplers. In Polaroid film, those chemicals are present within the film and therefore do not need to be added after the image is captured.
Bringing the Photo to Life
Instant camera film contains three layers that are each sensitive to a different color of light. Beneath each of the layers lies a developer layer that contains the dye couplers. All of the layers are stacked on a base layer, which is black. On top of the color layers are even more layers. Those are three in number and include the image, timing and acid layers. The arrangement of the many layers isn't a coincidence, but an important configuration of chemicals that will all work together in harmony the second you hit the button on your camera. The "spark" that sets it in motion is called the reagent, which is a mix of elements that include (but are not limited to) white pigment, acid neutralizers and light-blockers. Before the button on your Polaroid camera is pressed, all of the reagent material is gathered in the white frame of the soon-to-be photo. That keeps it away from all of the developing chemicals until the proper time, which prevents the film from exposing prematurely.
The Finish Line
After you've taken your photo, the camera expels it. It's partly that action that gets things going, as it were. The rollers that move the photo also move the reagent from the edges of the photo to the middle and, subsequently, down through the many layers of the photo. From there, the photo begins to develop. Everything has come together and slowly you'll see the magic begin to happen.