[Arwen Imladris]

'I am a servant of the Secret Fire'

This could be refuring to two things. One is Gandalf's elven ring, narya, the ring of fire. However it is more likely that he is refuring to the Flame Imperishable, the source of life that only Iluvatar could give.

'Wieldor of the flame of Anor. '

Anor means 'the Sun', and so literally the 'flame of Anor' would be the flame of the Sun. This could mean the fruit of Laurelin, one of the Two Trees of Valinor.

'The dark fire will not avail you, flame of Udun.'

This seems to simply be a name for balrog, however it could also be refuring to 'Udûn', Morgoth's oldest and greatest fortress in the north of the World, more commonly called Utumno.

Hope that helps.