The coloured belt system was made in the 19th century by Kano Jigoro who was the founder of judo. Before that they usually only had the white and black belt, once you had done so many years you would be handed a black belt.
There is a myth that goes round that the belt system of colours was inspired by the training with the white belt, since you would have years till you get your black belt the white belt would get dirtier going from lighter to darker colours. That's why the colours were like what they were to determine rank, the darker they were the more likely the more years you had done.
Systems that do not use coloured belts are more likely to be self defence martial arts e.g. krav maga.