Rainbows are the result of sunlight reflecting off of drops of water in the atmosphere. In order to see a rainbow, the sun must be behind you, and raindrops in front of you.

The sunlight enters the water droplet, and is separated into the different colors of the light spectrum. If you remember from school, the light spectrum is red, orange, yellow, green, blue and violet. Light is separated because different colors of the spectrum bend more than others within the drop of water. (Red light bends the least, violet light bends the most.)

The light is reflected off the back of the drop of water, and back towards the observer. We see red light reflecting off droplets at the top of the rainbow, and violet light reflecting off water droplets at the bottom of the rainbow.

The drops of water have to be at just the right angle for the light to reflect back to you, that's why you only see a thin strip of the sky reflecting light back at you.