Gandalf's staff, I think, functioned as a channel for his powers, and it was a symbol of his power. The principle applied for the rest of the Istari. In The Two Towers, when Saruman was thrown down by Gandalf, his staff was broken, symbolizing the breaking of his power. Also, when Saruman is insulting Gandalf, he says:
Quote:
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Later! Yes, when you also have the Keys of Barad-dur itself, I suppose; and the crowns of seven kings, and the rods of the Five Wizards....
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That must have meant that the power of the Wizards was "vested" in their staffs, and the taking (or breaking) of their staffs would result in a loss of power, like what happened to Saruman.