Rainbows are the result of the interaction between Rain droplets and Sunlight and these are one the most colourful events of nature. To see a Rainbow was has to have the sun behind you. The angle of the sun has to be 42° above the horizon and rain, Fog or some other source for the water droplets needs to be present. Raindrops are spherical and this allows the rainbow to be seen. The size of the rain drop doesn't affect the geography rainfall although fog or mist will disperse the effect more. To see the full semi circle of the rainbow one has to be either at sunset or sunrise and on level ground when the sun is on the horizon. When the sun is higher which will be the majority of the time only a smaller section of the arc will be seen.
However if you see the rainbow from an aeroplane or a mountain and the angle is correct you should see the full circle.
Sunshine has a spectrum of many colours and the reason why the sun looks white is that the colours are mixed. Light travels slower through water than it does for air so the light is therefore bent as it enters the raindrop and ends up being refracted. This splits the light into the colours of the spectrum and also some light is reflected from the back of the raindrop causing the colours to be reversed to provide the order of colours we are so familiar with.
The colours of the rainbow are Red, Orange, Yellow, Green, Blue, Indigo and Violet. At school you will have learnt that they are seven colours. In fact this isn't quite true and if you look closer they are actually far more. The number seven sticks to this day as it helpful to remember the colours of the rainbow however there is a wide range of colours not all visible due to the limitations of the human eye.
The photo above actually shows a double rainbow. These are formed when the sunlight is reflected twice within the raindrop. The colours of a double rainfall are reversed. The double rainbow is quite common and are more common when the sun is low early in the morning or in the late afternoon. The 2nd bow is always fainter and is dispersed over a wider section of the sky. The dark band between the two is called the Alexander's band after Alexander of Aphrodisias who first commented on it in 200AD
Moonbows are a rare cousin of the rainbow. These form when just like Rainbows but where the light is from the moon is reflected via raindrops in the air. Since the light from the Moon is much less than that from the sun is the reason why they are rare. The rarity is also due to the lack of colour receptors in the human eye. You more likely too see such effects in other parts of the world Hawaii being one such example.
The moonbow often appears to be white to the human eye however the full spectrum of the light is there. Using a camera with a long exposure such as the one by Ian Glendinning is excellent way of capturing their beauty.
Due to the lack of light the moon needs to be at it fullest phase and again low in the sky. The night sky needs to be very dark so this would rule cities or places near cities due to light pollution. Rain needs to be falling opposite the moon and the obsever needs to have the moon behind them. Moonbows are best viewed a few hours after sunset or sunrise.
Fogbows form when diffraction of the light occurs due to water droplets rather than rain drops within the fog, mist or cloud.
The size of the water drops are much smaller than raindrops mostly less than 0.1mm. The size of the water drops use a different process called diffraction and this leads to the bow to be devoid of colour so that they appear white. They are also known as white rainbows because of this.
Fogbows are often accompanied by another optical phenomenon known as Glory which also caused by diffraction.
To see a Fogbow you need the same basic parameters. Sunlight at the observers back and the water droplets in front. Since the droplets are so small when compared to raindrops the process of diffraction is the main effect. The process of diffraction broadens the reflected light which smears out the colours making the fogbow appear white or very faint fogbow. The way the light is reflected also means the fogbow is much broader than the it's cousin the Rainbow.
The fog also has to be fairly diffused and thin otherwise the light won't be allowed to pass through the droplets which create the effect. You may also see the same effect while flying and these are known as Cloud Bows.