The Secret Fire seems to refer to that aspect of Eru Ilúvatar which is his Power of Creation. This power was with Ilúvatar at the very beginning, so when Melkor sought for it alone it was in vain, for he could never find it because it was solely with Eru himself. Later, Eru set it (or an aspect of it) at the Heart of Arda.
|“||Therefore Ilúvatar 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 Eä.||”|
The Secret Fire was also described as being part of all Fëa, the soul of a living being or the spirit: it was that Gift which makes sentient beings capable of independent thought and will. Loosely, it could be taken as an alternate name of the Holy Spirit.
|“||You cannot pass," he said. The orcs stood still, and a dead 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 Udûn. Go back to the Shadow! You cannot pass.||”|
Anor was a name for the Sun in the mythology, as well as a name for the True West (i.e. Valinor). Further, some have speculated that Gandalf said "flame of Anor" in reference to the fact that he wielded one of the Three Rings of Power, Narya. Thus it is uncertain if the flame of Anor was identical to the Secret Fire, but both it and The Secret Fire identified Gandalf as a servant of the Valar.