Rainbows like the ones we see during or after rain come from white light of the sun interacting with tiny droplets of water in the air. During or after rain, there is a much higher number of small water droplets light enough to float in the air. The water droplets can reflect light, bend light, or disperse light - dispersing light generally means splitting light into its different colors. In the case of sunlight, because it is mostly white light, its dispersion by the water droplets can result in the full spectrum of visible light, which we perceive as a rainbow. Because of the way light works, we as observers can only see rainbows at certain angles relative to the source of the light. The required angle explains why, when we use a hose with fine spray of water on a sunny day, we can see small rainbows only from certain directions and not others.

Side note: We see distinct colors in a rainbow because our eyes are not sensitive enough to see every wavelength of the visible light spectrum. In reality, every wavelength of the visible spectrum is in a rainbow.

Rainbows happen when sunlight shines through rain drops. Sunlight is white, but white light contains all of the other colors. When white light shines into a raindrop, the light bends and is reflected off the inside of the drop and back out to the viewer. When this happens, each component color bends by a different amount, so we see them separately as a rainbow.

Rainbows are caused by sunlight reflecting off of the back sides of rain drops back at the viewer, but because different colors of light are bent different amounts as they enter and leave water, they are smeared out through the electromagnetic spectrum, creating bands of different colors as light from slightly different angles reaches your eye.

White light has all the colors of the rainbow. This seems a bit strange, because you'd get paint that was very dark if you mixed all the colors together.

White light from the sun can bend when it hits water droplets in the atmosphere. Different colors of light have longer and shorter wavelengths, and they bend more or less, so the light from the sun is separated into all the colors of the rainbow. You can see, from the rainbow, which colors of light have wavelengths that are the move similar, because those colors are next to each other in the rainbow.

Here's part of what Wikipedia says about rainbows:
A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicolored circular arc. Rainbows caused by sunlight always appear in the section of sky directly opposite the sun.