A rainbow comes upon us by surprise, most of the time. They are
not really rare, but they are short-lived, and you have to know
when and where to look.
Cumberland Falls, near Corbin and London, KY, provides a nice example of a rainbow at a waterfall. This waterfall faces north, and makes a lot of mist; in the morning there is a very vivid rainbow to be seen from the viewing area at the east side. Even more interesting is what happens in the early evening on days near the full moon: now the moon is rising in the east, and is bright enough to give a "moonbow." It is too dim for your color vision to work well, so that it seems more like a colorless (that is, white) arc. This phenomenon is quite rare -- it requires a lot of mist (to give lots of scattered light of the right kind), but not too much (which would just give you a white cloud to look at); it has to face the right way; and it's surely very important that Cumberland Falls is a state park, with no street lights behind the moonbow. For some great time-exposure photographs of the moonbow, see Mike Martin's Moonbow Museum.
Rainbows occur in nature when a beam of light interacts with many drops of water. Both reflection and refraction of the light play an important role in their formation. Many different things can happen to the light on its way through a typical drop, but most of these just give light redirected in all directions. The rainbow originates in the special case when a beam of light enters a drop (instead of being reflected away from the drop at its surface), is reflected once inside (instead of leaving at first opportunity through the back surface of the drop), and then finally leaves when it encounters the drop's surface for the third time.
(This picture comes from a simulation
by Sadahisa Kamikawa, which you might like to play with).
The direction of the scattered light depends on how the beam strikes the raindrop. The picture below shows a spherical drop encountering parallel beams of light. The important points to see in the picture are that all beams are sent back to the left, that many of them are going nearly the same direction (compare the directions of the beams that leave near the bottom left corner), and that none of them are scattered more widely than this. All of these beams are reflected more or less backwards, so that to see them we should be looking away from the sun; we might expect to see a bright glow in that direction. But the interesting feature is that most of the beams that hit the upper side of the drop are scattered in almost exactly the same direction -- about 42 o away from the direction back to the sun. This means that when you look at an array of many drops, you will only see the ones that are 42 o away from the shadow of our head. This defines a ring of light, centered on the point where the shadow of your head would be, and having a radius of 42 degrees. The size of this ring depends slightly on the color of the light, so that the "red" ring is inside the "blue" ring. Usually we can only see the part of the rainbow ring that is above the horizon, because we need lots of water drops in every direction that we will see it. We are told that from an airplane you can sometimes see the complete ring.
From this description, we can deduce the important conditions for seeing a rainbow.
Rainbows are to admire and enjoy -- you can't put them in a jar, and they are hard to photograph, because they are not really much brighter than the sky.