Exactly what power did the Ring of Fire give to Gandalf?

[Boromir88]

I don't think Narya being called 'The Ring of Fire' has anything to do with the power that it gives to Gandalf. It was also called 'The Red Ring' as the stone set in it was a ruby.

The Elven Rings weren't made to be weapons, or to increase their bearers powers. There powers were in preservation and protection. Galadriel sustains and protects Lorien by the power of her Ring, same as Elrond. But the Rings themselves are not actual weapons.

As far as Narya's powers, I think Cirdan offers a good explanation:

'Take this ring, Master,' he said 'for your labours will be heavy; but it will support you in the weariness that you have taken upon yourself. For this is the Ring of Fire, and with it you may rekindle hearts in a world that grows chill.'~Appendix B: Tale of Years; The Third Age

I think we see the power of Narya at work in a couple instances. One being in the battle with Durin's Bane. But not by giving Gandalf any sort of increased power, but by giving him strength when he was weary. As Cirdan says 'it will support you in the weariness that you have taken upon yourself'...Gandalf remarks that his confrontation with the Balrog at the gate in Balin's Tomb (although at the time Gandalf didn't know it was a Balrog), had drained him and he was already 'weary.'

Also in the Siege of Minas Tirith for some reason people were inspired to continue to fight when Gandalf was around:

So it was that Gandalf took command of the last defence of the City of Gondor. Wherever he came men's hearts would lift again, and the winged shadows pass from memory.~The Siege of Gondor

While Galadriel and Elrond used their rings to protect (as well as 'stop the decay of time' - preserve) their realms...Narya you could say 'preserved' the heart in men.