How does a a rainbow form?

Going through a prism, light is decomposed into every color in it.

The sunlight shining through the droplets of moisture in the Earth's atmosphere make the spectrum of light appear in a semi-circular form.

Keep in mind that the colors you can see is only the colors that can be seen by the human eye - infra-red and infra violet will be present in a rainbow but will not be visible to us.