Manifestation (顕現 Kengen ) is the materialization of Crimson Denizens in the human world and the New World Xanadu. It also refers to how the combination of someone's intentions, mental image, and the use of Power of Existence, can materialize substances and Unrestricted Spells.
Essentially, Denizens, Flame Haze and Torches all consume Power of Existence in order to manifest, though there are some exceptions. One example of this would be a Denizen that was on top of the Treasure Tool Caina; as long as it stayed on Caina, its manifestation would continue without consuming Power of Existence.
Another example is Dantalion, who had the unique ability of being able to materialize substances.
The Power of Existence used for the manifestation can be used to an extent that the user's body becomes only semi-manifested due to the lack of power. Tiriel is forced to resort to this when she didn't have enough Power of Existence to protect Sorath and stop Shana's attack.
As Denizens cannot materialize their true form in the human world, their essence (their own Power of Existence) is converted and manifested into another body, and all actions they conduct within the human world is done using this new body.
For Flame Haze, the strengthening of their body and the use of Unrestricted Spells is the manifestation consuming the Power of Existence of the Crimson Lord dormant inside them. While within their Flame Haze, Lords don't need to consume Power of Existence. For a Lord that has lost their contracted Flame Haze, they must either return to the Crimson Realm or use their own Power of Existence and consume it from others in order to remain in the human world, or else they will be extinguished.
In the case of a Torch, they consume their remaining Power of Existence and materialize a body as substitute for a human. However, if they have a strong will, and their remaining Flame is strong enough to withstand consumption, they can learn how to use Power of Existence and manifest a body that is beyond a substitute for a human, and even learn how to use Unrestricted Spells.
For Crimson Gods, it is the same as Denizens, except when they are summoned to the human world through a Divine Summoning. If this is the case then the manifestation of the God doesn't need to consume any Power of Existence from the human world or from themselves.