Technically speaking, colors are the way our brain, by use of our eyes, interprets electromagnetic radiation of a wavelenght within the visible spectrum. Visible light lies between 400 and 700 nanometers.
The different wavelengths are seen as different colors, as in the spectrum below. You see a spectrum like this everything you see a rainbow.
Traditionally the spectrum is divided into seven separate bands. The first person to really define this was Newton. Some people believe that he included the color indigo only to make seven steps to match the number of notes in major musical scale. These are the approximate wavelengths for each of the colors:
Electromagnetic radiation with a wavelength of over 750 nm is called infra-red, and radiation under 350 nm is ultra-violet.
Each color on the spectrum can vary in saturation, lightness and darkness, and it's estimated that a human eye can distinguis about 10 million color variations.
In color theory, we often talk about the color wheel. A color wheel is really just the spectrum twisted around so that the violet and red ends are joined. The color wheel is particularly useful for showing how the colors relate to each other and how you can create new colors by mixing two or more colors.