When you look at the rainbow, what you see is the prism like effect of the mist (aerosolized water droplets) in the air reflecting the sunlight from different portions of the spheres.  These water droplets when suspended in air as mist will all reflect different colors at different angles.  The angle between you, the mist and the sun, will then determine which color is being refracted back to you from each location resulting in a rainbow.  This color is itself a special form of radiation, more specifically it is non-ionizing radiation with very specific wavelengths.

Your eye is one of the most sophisticated radiation detectors ever designed with the ability to discriminate color in this way and allow it to be processed as it is.  Similar to light, gamma rays and radio waves are a form of electromagnetic radiation.  Radiation can have wavelengths that are very long such as radiowaves or ones that are incredibly small like gamma rays.  This distribution of wavelengths is called a spectrum.  The spectrum goes from wavelengths which can be as large as the earth to incredibly small wavelengths smaller than an atom. 

The eye can only see a very small portion of this spectrum. It is visible radiation that actually defines what we commonly call "light".

The electromagnetic spectrum goes all the way down to low energies below infrared and radiowaves all the way up past the high energies of x-rays, gamma rays and cosmic rays.  Somewhere near the middle is what we call the visible range of the electromagnetic spectrum (i.e., light).  What this means is that there are many forms of electromagnetism with visible light being only one of many forms with all these forms only differing by their wavelength. 

Light is the form of electromagnetic radiation having wavelengths around one half of one millionth of a meter in length, smaller than blood cells or even chromosomes (but much larger than atoms).  Our eyes are designed to discriminate between different wavelengths of electromagnetic radiation near this range, this small band of energy ranges is what we call color.

Just as gamma rays and radiowaves only differ by their wavelength, the same is true for color.  The higher energy visible light is violet.  Just above this in energy is ultraviolet which cannot be seen with the naked eye and just starts the range of electromagnetic waves that are ionizing and associated with cancer incidence at high doses (sunlight carries this kind of radiation along with visible light).  The lowest energy visible light is red with energies just below this being called infrared.  Infrared radiation is the type you can actually feel with your hand when sitting across from a bed of burning coals (heat radiation).

The light we see from objects is actually only the reflected light from the item.  If you shine white light on an object, that object will absorb some fraction of that light and reflect the rest.  The reflected portion is what we see and call the color of that object when in fact it is really the color not absorbed from the same. 

All natural light sources contain a mixture of colors so that when this light is run through a prism, the components can be split up and recognized.  It is these portions of the mix that reflect off of objects to give them their distinct colors, the portions absorbed by an item are simply converted to heat.

Some materials such as phospors can also shift the energy of reflected light to lower energies so that incoming light from the invisible ultraviolet range can be downshifted to the visible range upon reflection.  This is how a black light works.  You cannot see the direct ultraviolet light from the source but when it reflects off of certain substances such as white clothing, the reflected distribution has a lower energy in the visible range and can now be seen with the naked eye.  The light looks black but it is really emitting radiation at energies we cannot see but the reflected portion coming off of certain materials has a lower energy in the visible range giving the familiar "black light effect".