" 'You cannot pass,' he said. The orcs stood still and a silence fell. 'I am a servant of the Secret Fire, wielder of the Flame of Anor. You cannot pass. The dark fire will not avail you, flame of Udun. Go back to the Shadow. You cannot pass!' "Some people have taken this to mean that Gandalf serves his Ring of Fire, Narya. But this is not consistent or appropriate when Gandalf's history is taken as a whole.
[The Fellowship of the Ring]
I was re-reading the Silmarillion for references to what the "Secret Fire" that Gandalf serves, and I found the following:
"He [Melkor] had gone often alone into the void places seeking the Imperishable Flame; for desire grew hot within him to bring into Being things of his own, and it seemed to him that Iluvatar took no thought for the Void, and he was impatient of its emptiness. Yet he found not the Fire, for it is with Iluvatar."Then later:
"Therefore Iluvatar gave to their vision Being, and set it amid the Void, and the Secret Fire was sent to burn at the heart of the World; and it was called Ea."The Imperishable Flame and the Secret Fire seem to represent the same thing; the power of Iluvatar to impart actual Being to his thought - the spirit of creation, if you will. It seems fitting that this would be something that Gandalf (as a Maia sent from the West by the Valar) would "serve" as counterposed to the evil of the Balrog. In this context, any notion of Gandalf serving the power of his ring Narya is absurd, IMHO (g).
As to what is meant by Gandalf's words "Flame of Udun", I simply infer that this is another word for Balrog. Balrogs are demons of fire, and the word Udun is found as an entry in the glossary of The Silmarillion under "tum":
"Cf. Utumno, Sindarin Udun (Gandalf in Moria named the Balrog 'Flame of Udun'), a name afterwards used of the deep dale in Moria between the Morannon and the Isenmouthe."And Utumno is of course the first stronghold of Melkor in the North of Middle-earth. Hence, Flame of Udun could be read as Servant of Morgoth or Balrog from Morgoth's Fortress [Udun].
Just conjecture, of course.