The sun's light is made up of several colors ranging from red though green and into blue and violet. You see the sun's light as white because you're seeing all the colors combined.
But raindrops can act like a prism in separating the sun's light into its individual colors. Each individual color of light gets bent slightly different when it hits the raindrop, spreading out the colors into what you see as a rainbow.
Double rainbows occur when the sun's light gets refracted twice inside the same raindrop. Since some of the light has already escaped to make the first rainbow, double rainbows are generally dimmer. Their colors are also inverted.
For More Information: