A rainbow is produced when white light in our atmosphere is refracted by water droplets. White light is comprised of the colours red, orange, yellow, green, bue, indigo and violet, and these colours have different wavelengths, (red the biggest: 650nm, and violet the smallest: 400nm).

Refractions occurs because light waves travel slower through denser mediums, (transparent materials), therefore when a light wave enters a denser medium at an angle other than the normal, (adjacent), it changes direction. Hewever this direction change is greater the smaller the wavelength of the light. This means that violet light is 'bent' more than indigo light, indigo more than blue, blue more than green, etc.

The different direction changes of the light is the reason that the colours seperate forming the spectrum visible in a rainbow; with violet in the inner-arch through to red in the outer-arch. (This is the true order of colours, where your "lighter than the outside" idea came from i have no idea.) But i hope this has helped.

The answer about refraction is what you are taught in school, but is fairly naive and does not account for all the phenomena.

Think about one light ray hitting one water droplet. If it hits dead central, it goes straight through. As the hit point moves off-centre, it gets one refraction and then goes out the other side. There is a distance from the centre where a refracted ray meets the surface of the droplet the second time at a shallow enough angle to get completely internally reflected (i.e. there is no external angle that corresponds to the refraction of the internal angle).

Depending ONLY on the distance off-centre of the original ray, a single ray can get three or four internal reflections and then manage to escape at one of two very narrow angles of dispersion. Less than 3 hits, it emerges near to its original direction. More than 4 hits, it never attains an escape angle. Three and four hits give us the primary and secondary rainbows. Because a droplet is spherical and homogeneous, different sized raindrops do not affect any of this.

Also, this whole thing is in 3 dimensions. Visualise an open umbrella. Imagine a lot of rays going parallel from the handle end hitting a droplet at the spike end of the umbrella. Because the droplet is round, the rays that exit splat out towards you all round, like the spokes of the umbrella. You only see the ones that keep a constant angle between you and the Sun, because these reinforce each other. Other people in different places see other rays from the same drops coming to them at a different angle.

A rainbow is not in a specific place, it is a direction: if there is a great depth to the curtain of rain, the bow is more intense. Also, if it is raining hard you can often see the rainbow in front of things like trees in the distance.

The sky region inside a rainbow is indeed lighter than the sky outside. The rainbow "steals" some light that would normally carry straight on through the rain. Some of it is carefully arranged into a rainbow for you. The rays that come out (from your point of view) in directions that do not reinforce in colours, recombine into white light. This is what brightens the sky within the rainbow.

Incidentally, I saw a straight rainbow once - vertically up into the sky! This was near to sunset at the eastern end of Gairloch in Wester Ross, Scotland. You know when the Sun sets over water, you see a "golden river of light" because the ripples catch the light? This effectively makes two Suns - one point source and one extended line. With my back to that, and a rain squall over the hill in front, there was a regular double rainbow, and then a separate straight bow caused by some reinforcement effect from the extended straight light source. Unforgettable sight!

Further to my answer on 18-Dec-2009: I sat down to figure the exact angles and so on in a droplet, but while I was researching Snell's Law and refractive indexes, I found a superb explanation in a Forum - much better than anything I can do. All credit to Fu-Kwun Hwang and the National Taiwan Normal University. Small points:

(1) It's a Java applet, so you will need a Java RunTime Environment installed. It takes a while to initiate and close down. The diagram is half a page down the first forum posting.

(2) You can click on the small red/green/blue/white boxes to select the incident light colour. It shows you the refractions/reflections, and the intensity distribution of the resulting colour output from 0-360 degrees.

(3) It shows you one incident ray, and the primary and secondary results come out on opposite sides of the diagram. What happens for real is that the rays making the primary rainbow you see go in the top half of the droplet and come out the bottom, and the rays making the secondary go in the lower half of the droplet, go round in the opposite direction, and come out the bottom too. That is, the whole diagram can be rotated about the horizontal axis. Click on the 4-coloured key square to see this.

(4) My initial explanation was way too simple. The sun's light is polarised. There are also quantum effects. So each ray hitting a surface both reflects and refracts - there is no such thing as "total internal reflection". The proportion of the intensities of the split at each surface is determined by phase angles etc - in fact, the refractive index is a complex number (real and imaginary parts).

The link to the page containing the Forum with the java applet is is:

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=44

The light you see in the daytime is coming from the sun, obviously, and if the sun is behind you then the light you see is caused by scattering from the atmosphere. When you see a rainbow you see light from the sun, which must be behind you, refracting and reflecting of the inside of raindrops. The light enters one side of the rainbow and is refracted into a spectrum. Most of the light continues out the other side of the rainbow but some is reflected from the other side of the drop and returns to you. Different wavelengths of light are refracted different amounts, and the amount that they are refracted determines at which angle they achieve 'total internal reflection', so that the point at which the different colours return to you is a function of the angle between you and the antipode of the sun. So a rainbow forms as a circle in the sky with it's center directly opposite the point of the sky in which the sun is located, from your perspective. The double rainbow is caused by a second set of reflections inside the raindrop. Meanwhile, the light from the sun that has been refracted and reflected inside the rainbow does not go on to scatter off the atmosphere ahead of you as it would have if there were no rain, so the rainbow has effectively removed light from the arc of the sky behind it. That's what causes the dark band between the rainbows, called 'Alexander's Dark Band', named after Alexander of Aphrodisias who first commented on it in 200 AD.

You can get more than a double rainbow; New Scientist reported the first photograph of a quadruple bow in October 2011. (http://www.newscientist.com/blogs/shortsharpscience/2011/10/third-and-fourth-order-rainbow.html). This image of this quadruple bow has been doing the rounds on the internet (http://www.jokeroo.com/pictures/nature/quadruple-rainbow.html, can't find the original source). It's not a true quadruple bow, but two double bows, it appears that one was caused by the Sun and one caused by the reflection of the Sun in the lake behind.