During the LOTR we see Gandalf change from 'Gandalf the Grey' to 'Gandalf the White' and we see 'Saruman the White' change to 'Saruman the Many-coloured'.
What does this mean and how and why does it happen?
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Gandalf and Saruman discuss Saruman's change in the Fellowship of the Ring:
"I am Saruman the Wise, Saruman Ring-maker, Saruman of Many Colours!"
'I looked then and saw that his robes, which had seemed white, were not so, but were woven of all colours. and if he moved they shimmered and changed hue so that the eye was bewildered.
' "I liked white better," I said.
' "White! " he sneered. "It serves as a beginning. White cloth may be dyed. The white page can be overwritten; and the white light can be broken."
' "In which case it is no longer white," said I. "And he that breaks a thing to find out what it is has left the path of wisdom."
Saruman views white, as a colour, to not be sufficiently demonstrative of his now-revealed power - he considers his Ring making, Uruk-hai creation and army raising to be the work of someone "more" than a mere servant of the Valar, as Saruman the White was once. Gandalf reflects on that in his last comment - "he that breaks a thing to find out what it is has left the path of wisdom".
Gandalf, on the other hand, embraced his change in colour to white:
'Yes, I am white now,' said Gandalf. 'Indeed I am Saruman, one might almost say, Saruman as he should have been. (The Two Towers)
He has died and been sent back by Eru to finish his task. Since Saruman has passed off the path of the Istari, Gandalf is sent back with greater powers and a change in colour to reflect that. He now, in effect has similar powers to, if not greater than, Saruman:
both his wisdom and power are much greater. When he speaks he commands attention; the old Gandalf could not have dealt so with Theoden, nor with Saruman. (Letter 156)
Gandalf's change of colour is essentially to signify his position as the Saruman that should have been, with Saruman's fall.
So as to "how" a Wizard changes his colour, it could be self-declared as per Saruman, though doing so would tend to indicate deviation from their assignment. Alternatively it could be granted by a higher power, as per Gandalf. It obviously differs in the only two cases we see.
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.
The color of the robes of an Istari can possibly change by gaining power. It could also be changed by the wizard, as shown by Saruman when he became "Saruman of the Many Colors". However, this may not be a true change of color. It is unknown how the color of a wizard truly changes, but as the wizard grows more powerful, he may change his color.