Have you ever seen a "sundog"? It's kind of like a rainbow, except it's formed by ice crystals, not water droplets. Sundogs form when the sun's rays pass through the ice crystals and are refracted at certain angles. (Hang on, this is going somewhere...)
Depending on the shape of the ice crystal (cylinder, plate, flake etc...) that the light is passing through, the angle of the refraction will change. (If the angle of refraction is small - the arc will be near the sun, if the angle is big - arc will be a big circle around the sun.)
Now back to rainbows... Water refracts light in at much larger angle than ice (Biiig circle...). Also, you most often see a rainbow when the sun is shining through falling rain (duh.) - when it (the sun) is lower in the sky.
Both of these add up to a very big angle "sundog" that is around a sun that is low in the sky. This causes the majority of the rainbow to be "below the horizon" - so you only see a part of the circle (an arc).
BTW - if you've never seen a sundog, try on a day when there are a few cirrus clouds in the sky. These are really high in the atmosphere and are made of ice. Sometimes the circle is very close to the sun - only a few sunwidths wide. Sometimes they seem to take up most of the sky. Sometimes these wider ones look like rainbows, the colors aren't as brilliant, but they can still be distinguished.
The same thing happens with the moon (moondogs).