The Secret Fire and Flame Imperishable as well as possibly the Flame of Anor are references to a mysterious power.
The Secret Fire, also called the Flame Imperishable, seems to refer to that aspect of Eru Ilúvatar which is his Power of Creation. It was with Ilúvatar at the beginning, so Melkor sought for it in vain, and later Ilúvatar set it (or an aspect of it) at the centre 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. — Valaquenta, the Silmarillion.
The Secret Fire is also described as being part of all feä: it is that Gift which makes sentient beings capable of independent thought.
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 Fellowship of the Ring II 5: The Bridge of Khazad-dûm.
Anor is 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 is identical to the Secret Fire, but both it and The Secret Fire identified Gandalf as a servant of the Valar.