From AMS Glossary
Any one of a family of large, colored, circular (or nearly circular) arcs formed by light (usually sunlight) falling on a population of water drops such as provided by rain, cloud, fog, or spray.
The apparent center of the arcs is normally the shadow of the observer's head, so the rainbow is a personal phenomenon with each person seeing a slightly different bow. Although the rainbow can form a circle, when caused by rain the bottom part of the circle is usually cut off by the ground, leaving an arc, the extent of which depends upon the elevation of the light. The term rainbow is not applied to the small, nearly circular arcs seen around the sun and antisolar point in clouds. These are the corona and glory. Nor is the term applied to the large circular arcs formed by light falling on ice crystals. These are the halos. Rainbows are seen as related groups of arcs, such as the primary rainbow (with red on the outside, blue on the inside), the (larger) secondary rainbow (with red on the inside), the supernumerary rainbows (seen to the inside of the primary bow), and the reflection bows (the centers of which are above the horizon). However, the appearance of these bows can vary markedly depending upon whether they formed in rain, drizzle, or cloud, as both the radius and color purity of the arcs depend on drop size. Certainly, the brightest and most frequently seen of the bows is the primary rainbow, but whether the whole arc above the horizon is seen or not depends upon the location of the rain. It is not uncommon for someone to report seeing two rainbows, when these were merely two unconnected portions of the same bow. There is a hierarchy of theories of the rainbow with simplicity being purchased at the expense of verisimilitude. Theories that treat light as a series of rays do a good job of explaining the approximate positions and colors of the primary and secondary bows but fail to account for the supernumerary bows. To account for such easily observable features of the natural bow as the variations of color purity, brightness, and distribution of the supernumeraries around the arc, such things as the wave nature of light, the drop-size distribution in a cloud, and the size- dependent shape of raindrops must be taken into account.