Here is where Tolkien mentions the secret fire the first time in The Silmarillion(Near the end of the fourth paragraph of the "Ainulindale":
Never since have the Ainur made any music like to this music, thought it has been said that a greater still shall be made before Iluvatar by the choirs of the Ainur and the Children of Iluvatar after the end of days. Then the themes of Iluvatar shall be played aright, and take Being in the moment of their utterance, for all shall then understand fully his intent in their part, and each shall know the comprehension of each, and Iluvatar shall give to their thoughts the secret fire, being well pleased.
So it seems that the secret fire has something to do with knowledge, which is also one of the ministries of the Holy Spirit.
Wielder of the Flame of Anor is interesting as well. I am not so sure if it refers to the Ring of Fire or not. Anor was another name for Minas Anor (Tower of the Sun), which was changed to Minas Tirith after the Nazgul had taken over Minas Ithil and changed it to Minas Morgul.
To bear Anar, "Arien the maiden was...chosen because she had not feared the heats of Laurelin [the name of one of the trees of light], and was unhurt by them, being from the beginning a spirit of fire. Too bright were the eyes of Arien for even the Eldar to look on, and leaving Valinor she forsook the form and rainment...and she was as a naked flame, terrible in the fullness of her splendour.... Then Anar arose in glory, and the first dawn of the Sun was like a great fire upon the towers of the Pelori: the clouds of Middle-earth were kindled, and there was heard the sound of many waterfalls." - The Silmarillion Of the Sun and Moon and the Hiding of Valinor
So the light of the world was struck down by evil, but was lifted up in glory, by a spirit of fire, to give light to the world once more. Such a wonderful theme to have as a backdrop for such wonderful stories. Tolkien wished to avoid allegory, seeking 'application' instead. He was immeasurably successful.
"Therefore I say: Ea! Let these things Be! And I will send forth into the Void the Flame Imperishable, and it shall be at the heart of the World, and the World shall Be; and those of you that will may go down into it.' And suddenly the Ainur saw afar off a light, as it were a cloud with a living heart of flame; and they knew this was no vision only, but that Iluvatar had made a new thing: Ea, the World that Is." -The Silmarillion, Ainulindale
"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 Silmarillion, Valaquenta
I'll try to find the references, but the greater part of Melkor's evil was driven by a desire to posess the fire and make creations of his own. Ultimately he could only twist and pervert what others made.