This section applies to the ILCOE.
The concept of power is pretty central to the Quest for the Legends, being integral to the main plot and often discussed. However, it's an awfully vague term by itself, and while the details don't in themselves have much bearing on the story, I might as well shed some light on just what power actually is in the Quest for the Legends universe.
Fundamentally, QftL "power" is simply the thing that allows Pokémon to do all the seemingly supernatural things they do. Rather than being some vague relative description, it is an actual physical quantity - a rather abstract one, but still a well-defined single phenomenon. Think of it as a currency of sorts: you have some set amount of power, and then there are various feats you use that power to achieve. Different Pokémon possess both different physical amounts of power and have different means of using the power available to them, but fundamentally, this same quantity of power is at work whether you're Mew levitating, Bulbasaur conjuring leaves out of thin air, Entei creating a soul gem, or Arceus causing the Big Bang - the difference is just the amount that each one requires and which Pokémon are capable of using their power in that way. Legendary power is the same thing as ordinary Pokémon's power, but they have more of it and frequently more means of using it than other Pokémon.
Most actual Pokémon moves employ power to achieve things that would otherwise be impossible or extremely difficult, though Pokémon can also fight without power using simple physical force (the Mystery Dungeon games' standard attack that doesn't use PP would be an example of that). Power is also involved in Pokémon's quick healing and resilience. Humans generally don't have power, though human psychics and so on do exist and use this same abstract quantity of power to do what they do.
After power has been 'spent', it is recovered naturally. Under normal circumstances, the recovery is fairly quick and complete, i.e. the Pokémon returns easily to the same quantity of power it had before, with no permanent loss or adverse effects. On the other hand, if a very large amount of power is used, this process is both simply slower and needs to recover more power for a full recovery, so the maximum recovery time can be quite long. Food and drink, as well as manmade healing items, can speed up the process considerably, however.
Power can be lent and borrowed, but as Chaletwo mentions in chapter 52, it is always partly tied to the true "owner"'s soul, in the sense that as long as the soul remains, it has a limited amount of control over that power. This allows a Pokémon that has given another power to take it back without physically being close to the current bearer of that power. A body is always needed to actually do anything with power, however, and normally when a body dies, all power stored in that body is lost - hence the need for Entei and Suicune to transfer most of their power to other Pokémon first in order for the soul gem plan to work.
What the Destroyer does is to slowly drain power from the legendary Pokémon and make it his own (meaning that once lost, that power is no longer tied to the soul of the legendary it originated from). The speed at which this happens is dependent on the "magnetism" between the Destroyer and the legendary in question, which in turn can be expressed as the legendary's current power level multiplied by the Destroyer's current power level: essentially, if you're mathematically inclined, it can be approximated by the formula L'(t) = -aL(t)D(t) where a is some unknown factor. If you're not, that basically means it starts out slow, then gets faster as the Destroyer becomes more powerful, then slows down again as the legendaries become comparatively weak.
If you are mathematically inclined you might now point out that if this is correct their power level never quite reaches zero, but by the time a thousand years are up it's still for all intents and purposes nonexistent, and at this point the Destroyer sends out a pulse of power containing double the total power all the legendaries originally had. This power is distributed evenly among all of the legendaries it finds - meaning they are all just about equal during the War - and the sudden rush of power drives them into the madness that causes them to go on a wild rampage trying to kill each other. When the legendaries die in the War, their power isn't simply lost, as it usually would, but instead somehow lingers behind and gathers in the final legendary, as well as granting it the power of the Creator to create life out of nothing ("power" is a somewhat misleading word here; it's not a subcategory of the QftL power being discussed here, but more like an ordinary move that merely makes use of power). Then the cycle begins again.
In addition to draining the power of the legendaries, the Destroyer also inhibits their normal power recovery so that any power that is spent cannot properly be recovered: the legendary's power level goes down permanently rather than temporarily, hence the deal with Molzapart and Chaletwo trying to conserve their power.