Saruman changed his name to "of many colors" because he went crazy, it wasn't an actual title. Gandalf became "white" to take Saruman's place after he defected. A post he originally declined.
The colors do not seem to mean anything except for Saruman - as the head of the White Council, he wore White. Alternatively, it may have been called the White Council because he (its head), wore white. There is debate among fans as to which is correct. Gandalf was called the Grey because "he was clad in grey", even before coming to Middle-Earth. Radagast was the Brown, because he wore brown, possibly because of his love of nature. But their colors seem to be tied to their preferred wardrobe color. Presumably they could change their color at will if they really wanted.
The Maia spirits are sent of the Valar into the world. So when Gandalf "died" he was sent again of the Valar, with a new title, the one left empty by the defected Saruman.
