Why does my eyes see a red spot when over exposed to light?
You are describing the phenomenon of negative afterimages, a type of physiological afterimage, which is a result of how the cells in the retina work. Specifically, it is a result of the adaptation of photoreceptors.


  According to the opponent process theory of color vision, our
  perception of color is controlled by two opposing systems: a
  magenta-green system and a blue-yellow system. For example, the color
  red serves as an antagonistic to the color green so that when you
  stare too long at a magenta image you will then see a green
  afterimage. The magenta color fatigues the magenta photoreceptors so
  that they produce a weaker signal. Since magenta's opposing color is
  green, we then interpret the afterimage as green.


There is a well-known illusion, called the negative photo illusion, which exploits exactly that. It is required not to move the eyes, because you need to bleach the photoreceptors in an exact place on your retina. This is why you have to fixate on a dot in the middle of a picture, otherwise the bleaching will not occur.

EDIT: I mention a two-axis (magenta-green system, blue-yellow) system, but the exact color system - which would predict which afterimage you would see - is most likely a trichromatic system, which includes red, green and blue channels, based on the principle of the three types of retinal cone cells. It is important to remember that this is a phenomenon in perception, and thus may, in some cases, not be reducible to the fact that there are three types of cone cells, or that bleaching of a photoreceptor has occurred. Please see the comment thread below for some clarification.