Rayleigh scattering of the sun's rays occurs when sunlight strikes air molecules. Higher frequencies of light (green, blue, violet) are more readily scattered than lower ones (red, orange, yellow) so when we look at the sky away from the Sun, most of what we see is scattered blue light. The interaction of sunlight with much larger water droplets is categorically different. Instead of scattering, light traveling from air to water (or for that matter, across the boundary of any two different media) is refracted.
This means that the angle of the light ray to the normal (the perpendicular to the boundary between media) changes as it passes from one to another. The origin of this effect is the fact that light travels at different speeds through different media. The extent to which this occurs for different substances is measured by a medium's index of refraction, often denoted n. If two media have indices of refraction n1 and n2 then the angles of the light rays to the normal within each (denoted θ1 and θ2) are given by Snell's Law:
n1sinθ1 = n2sinθ2
For air and water, the indices take values nair = 1.000293 and nwater = 1.330. Snell's Law then yields the fact that light rays bend toward the normal as they pass from air to water and do the opposite upon exiting. However, these values of the indices of refraction are for a specific wavelength of light (actually a standard color of yellow light emitted from excited atoms of sodium with a wavelength of 589.3 nm). The degree of refraction varies slightly across the visible wavelengths, leading to the separation of colors that we observe as a rainbow. The small droplets of water in the atmosphere are roughly spheres, leading to the kind of refraction illustrated below:
Note that the angles by which the light rays are refracted depends on where it hits the drop (the redness of the lines has no significance in this image) since the boundary between water and air is spherical, rather than flat. Each of the rays shown undergoes a single internal reflection before emerging from the water droplet, though some light just passes through, and some is internally reflected multiple times (more on this later). However, the maximum angle between the incoming and outgoing rays are different for different colors of light: in particular, they are greater for longer wavelengths than shorter. Therefore, at the very highest angles, the colors are separated.
At one end of the spectrum, violet light has a maximum angle of 40° from the incoming light ray, while in the longest visible wavelengths, red light has a maximum angle of 42° (left). As a result, for a fixed observer, red light will appear to come from a certain angle in the sky, while violet will appear to come from another (right). Orange, yellow, green, blue, and indigo will appear in between. The result is what we see as a rainbow.
Several properties of rainbows follow directly from this understanding. The first is that all (primary) rainbows are of the same angular size in the sky, namely 42° in radius. A rainbow therefore does not have a fixed position and appears the same size to every observer, meaning that every observer in fact sees their own rainbow. Also, the center of the rainbow's circular arc must be opposite to the position of the Sun in the sky. This point is called the anti-solar point and must always be below the horizon (since the Sun is above). As a result, the higher the Sun is in the sky, the lower the (primary rainbow). If the Sun is more than 42° above the horizon, it cannot be seen at all. This is why rainbows are typically seen early in the morning or later in the afternoon. In addition, though the maximum angle is 40-42° for different colors of light, some light (of all colors) is reflected from raindrops at smaller angles, making the sky just inside the rainbow noticeably brighter. This effect is apparent in the image above.
Though most light reflected within the raindrop undergoes only a single internal reflection, some is in fact reflected more than once, leading to what are known as higher-order rainbows, notably the secondary rainbow.
The colors of the secondary rainbow are reversed since an additional reflection inside the drop reverses the color spread. Further, it is situated at 52°, outside the primary rainbow, and is considerably fainter.
The secondary rainbow is sometimes too faint to be visible, but it is always there. In fact, light can reflect internally even more, producing higher-order rainbows. However, three reflections sends the light on a path at about 43° inclined from its original trajectory, meaning that it would form a circle of this radius around the Sun. Due to its faintness and proximity to the Sun, it is very difficult to photograph, but photographs have recently captured this phenomenon (see below).
Thus, a simple application of atmospheric optics may explain the rainbow, the beauty of which has captivated humanity since antiquity.
Sources: http://www.atoptics.co.uk/rainbows/primary.htm, http://www.physicsclassroom.com/class/refrn/Lesson-4/Rainbow-Formation, http://0.tqn.com/d/weather/1/S/j/T/-/-/water-drop-prism-lrg_nasascijinks.png, http://ephy.in/wp-content/uploads/2014/11/clip_image0061.png, http://www.nilvalls.com/supernumerary-rainbow/, http://www.atoptics.co.uk/rainbows/ord34.htm