What causes a rainbow?

A rainbow forms when sunlight passes through raindrops and gets split into its component colors. White sunlight is actually a mix of all colors, and when a ray enters a spherical raindrop, it bends (refracts) as it goes in, reflects off the back inside surface of the drop, and bends again as it exits. Because each color of light bends by a slightly different amount (red bends least, violet most), the drop separates the white light into the spectrum of colors, red, orange, yellow, green, blue, indigo, violet, and sends them back toward you at slightly different angles. Millions of raindrops each doing this, at the right geometry, combine into the arc of colored bands we see. This is why you need both sunlight and water droplets (rain, mist, spray, or even a garden hose or waterfall) at the same time.

The geometry explains the rainbow's quirks. You will only see a rainbow when the sun is behind you and the rain is in front of you, because the light has to bounce back out of the drops toward your eyes at a specific angle (about 42 degrees). This also means a rainbow has no fixed location, it is not a physical object sitting in a place, but an optical effect that depends on your position, so everyone sees their own slightly different rainbow, and you can never reach the end of one (no pot of gold, sadly). A rainbow is actually a full circle of light; we usually see only the top arc because the ground cuts off the bottom half, but from an airplane you can sometimes see the complete circular rainbow. Sometimes a fainter secondary rainbow appears outside the main one, with its colors reversed, caused by light reflecting twice inside the drops. So a rainbow is sunlight refracted, reflected, and split by countless raindrops, a spectrum painted in the sky by the geometry of light and water, and personalized to wherever you happen to be standing.