What is the aikido colored belt system?

In Japan you are a white belt until you are black. Some Dojo out of Japan still follow that system however that's up to the Sensei. The multi color ranking system did not come from Japan, I believe the US started that but don't quote me on that one. Also in Japan they had purple belts. If you were not an adult they would not promote you to black, so you wore a purple until an adult.

The colors vary from organization to organization.

Our Dojo is white, yellow, orange, blue, green, brown, purple, black. That does not mean that another school might have them in a different order. The belts don't really mean anything except your understanding of the techniques.

Sometimes when you goto another Dojo, you wear a white belt, even if you are a black belt or brown belt. That's just being respectful. Usually the Sensei will give you permission to wear your colored belt. That's just the traditional way of doing it.