Above answers including explanations for "Secret Fire" and "Flame of Udûn" are satisfactory and correct as far as I know, but the treatment of "Flame of Anor" is massively speculative with no acceptable evidence, and thus woefully lacking. I will attempt to answer what Gandalf quite obviously meant by "Flame of Anor," using a style of exaggerated introduction to an enemy that Tolkien uses when Bilbo met Smaug that everyone seems to have missed.
The Flame of Anor is simply the Sun, and nothing else. More specifically, it is the Dawn. Gandalf meant quite literally that he wields the Sun, and we know this from his history to be true, from The Hobbit:
'Dawn take you all, and be stone to you!' said a voice that sounded like William's. But it wasn't. For just at that moment the light came over the hill, and there was a mighty twitter in the branches. William never spoke for he stood turned to stone as he stooped; and Bert and Tom were stuck like rocks as they looked at him. And there they stand to this day, all alone, unless the birds perch on them; for trolls... must be underground before dawn, or they go back to the stuff of the mountains they are made of, and never move again....
'Excellent!' said Gandalf, as he stepped from behind a tree, and helped Bilbo to climb down out of a thorn-bush. Then Bilbo understood. It was the wizard's voice that had kept the trolls bickering and quarrelling, until the light came and made an end of them.
Gandalf wields the Sun often subsequent to his battle with the Balrog. When reunited with Aragorn, Legolas and Gimli in Fangorn, he chooses to do so while bathed in the sun, from The Two Towers:
They all gazed at him. His hair was white as snow in the sunshine, and gleaming white was his robe; the eyes under his deep brows were bright, piercing as the rays of the sun; power was in his hand.
Arriving at Helm's Deep with reinforcements, we read the enemy is driven mad by Gandalf's approach from the East at sunrise, from The Two Towers:
There suddenly upon a ridge appeared a rider, clad in white, shining in the rising sun. [...] The White Rider was upon them, and the terror of his coming filled the enemy with madness.
Gandalf's use of sunrise is hardly isolated, and there are quite a few passages with him arriving at his destination at first light, from The Return of the King:
So Gandalf and Peregrin rode to the Great Gate of the Men of Gondor at the rising of the sun, and its iron doors rolled back before them.
In fact, Gandalf never reveals Narya until the very end of the story after Aragorn has been crowned, keeping it secret from the Enemy as instructed by Círdan, so its highly unlikely he would tell the Balrog anything of Narya. What Tolkien didn't reveal to us he never intended, such as Gandalf referring to Narya as "The Sun," which he would never do. Narya has no connection to the Sun other than the coincidence that Gandalf wields the sun, as he says, which is always a white light, and he also happens to secretly posses Narya, the red ring, giving him some influence over fire and the ability to kindle hearts. The Sun and the powers of Narya are not related. Narya is not a weapon nor does Gandalf use it as one, like he uses the Sun.