History
Coloured belts are a Western invention designed to denote progress and achievement. Back in Japan and China, there was no coloured belt system. A belt served a number of purposes - it kept your jacket closed so that it didn't get in the way, and it in the process, the jacket kept you warmer.
Modern martial arts has derived from Shaolin monks who lived in temples in ancient China. They developed a martial tradition that was at first voluntarily, then later compulsorily used in the service of the Emperor. The monks practised Buddhism and they lived a simple lifestyle, and had no need to impress people with their martial arts experience, so it's likely that they all wore simple saffron-coloured robes, and belts of the same colour.
At some time, martial arts instructors developed the habit of never washing their belts, and thus the longer one had been training, the dirtier the belt became, resulting in a black belt becoming a symbol of great experience or a very dirty lifestyle.
In any case, the belt has gained spiritual significance to many martial artists who say that the soul of their training and martial arts is in their belt, which has accompanied them throughout. Such martial artists continue to recommend that one should never wash one's belt, and whilst I respect their right to make this decision, it's nothing more than an affectation based on a tradition that had practical significance when it originated.
Given that we have dispensed with the original meaning of a black belt by introducing coloured grades,
Sometime after the martial arts moved to the West, instructors adapted the idea of the belt, to use it to denote experience. Modern Westerners felt the need for more achievable goals than mere experience, and they also wanted to denote smaller increments in progress.
In the West, there were originally two belt colours - White - a novice, and black - one who had finished his "apprenticeship". Then developed the habit of dying the belt to show progress. Initially, the same belt was simply dyed progressively darker colours: for instance:- white, yellow, green, blue, brown, black. Other colours include purple and red, although the grade of these belts varies from style to style.
Nowadays, nobody can be bothered to get the dye out each time you grade. You just buy a new belt from your sensei, so the colour progression is less important, and is no longer influenced by what colour can be modified into what colour - hence the fact that many styles now include striped belts.
