I find power level to be extremely confusing. (Frankly, before doing in-game research and learning more about how it worked I thought it was a systems blunder by Obsidian due to how obtuse and confusing it is.) I'm posting this share my research but also gather comments on power level, because I can't find an updated recent thread that isn't locked to talk about it. I only have incomplete information from staring at combat logs and a limited set of spells/abilities, so others feel free to chime in.
Power level scaling: affects all active abilities (including consumables). Weapon-based martial abilities get minimal and special power-level scaling (including special treatment by Empower, see end of post). Because weapon-based martial abilities are so special, this post will mostly focus on scaling done to other abilities. Note: this wasn't clear to me before, but adding this here -> all damage/duration bonuses you get from power level scaling are multiplicative with any other modifiers: they are applied first, and then all other modifiers are applied.
What it does: for every power level you have that is greater than the native power level of the spell, you get scaling bonuses. There are general rules about how spells should scale that I've found, though there appear to be exceptions. But what you see here should suffice for the vast majority of cases.
 some classes get spells from other classes' skill trees. For example, if you have Xoti train as a priest or priest/monk, at PL3 she gets a bonus spell: "Vile Thorns." Vile Thorns is natively from the druid skill tree, and so it is treated as PL1 in terms of power level scaling. So that means at PL3 you get access to Vile Thorns and it already has some bonus damage, even though you don't have a power level higher than the power level Vile Thorns is on, because natively it is a PL1 spell, not a PL3 spell.
General guidelines: First of all, tool-tip accuracy is inaccurate and inconsistent. The one you see when you right-click on an ability ignores ability and power level scaling. The one you see when you hover over the ability in your HUD ignores power level scaling, but counts ability level scaling. Anyway, regardless of power level, there is a scaling ability level accuracy bonus, which is equal to 2 * (power_level - 1) of the spell. So a PL3 spell will inherently have a +4 ability level accuracy bonus, whereas a PL1 spell will have none. There is also a scaling ability level penetration bonus, which is equal to +.5 per ability level.
Second of all, the general way scaling appears to happen is, first, take the difference between your current power level and the spell's native power level. For simplicity's sake, let's just call this the "PL". (So casting a PL1 spell at PL4 you would have a PL of 3 for scaling purposes.)
A. if a spell bounces or has projectiles, it gets an additional bounce or projectile every other PL. Spells used to have variable projectile scaling but it looks like that got nerfed at some point and they appear to get .5 projectile per PL.
B. if a spell does damage/healing, it gets +5% per PL. Non-bounce, non-projectile damage/heals used to get up to 10%, but looks like that was nerfed at some point.
C. if the spell has duration effects, it gets a longer duration of +5% per PL.
D. if the spell has penetration, it gets an additional +.25 penetration per PL (rounds up to the nearest tenth).
E1. if the spell has an accuracy roll, it gets +1 accuracy per PL.
E2. if the spell primarily only has that accuracy roll (no damage/healing, no duration effect), it instead gets a +2 accuracy per PL. I'm not actually sure how many of these types of spells exist, but I noticed this while playing with Repulsing Seal (which only does a prone). Slicken is another example of a prone-only thing (though it also has a hazard duration, a hazard duration must not prevent a spell from getting +2 accuracy per PL).
Empowering a spell gets you +5 PL to that spell when you cast it.
Because some spells have multiple components to it that may touch on A, B, C, D, E some spells disproportionately benefit from power level scaling than others. Anecdotally, empowering a Minoletta's Concussive Missiles can be extremely powerful, because it is almost getting a boost in everything: damage, projectiles, accuracy, penetration. On the other hand, empowering a different PL4 spell like Form of the Delemgan will certainly give me a longer-lasting buff, but is not going to single-handedly swing a fight like empowering concussive missiles.
Martial abilities: Non-weapon based seem to roughly follow the spell rules, but weapon-based (primary attack or full attack) abilities follow a special set of scaling, being only affected by B and secretly get an adjustment to their damage in A, but applied as a direct adjustment to the base damage roll, making it effectively a secret multiplicative damage boost. Because it is secretly applied, it's kind of hard to suss out what the bonus is, but my best guess from lots of force-attacking Eder is +5% to your roll per PL. Interestingly, doing an Empower-ed weapon ability manually adds a special damage, accuracy, and penetration bonus: +25% damage (additively combined with any inherent PL scaling), +10 accuracy and +2.5 penetration. I call this a special bonus because in the combat log, this accuracy/penetration bonus isn't attributed to power level scaling at all, it's attributed to the ability itself. And like normal power-level scaling, the Empowered damage bonus isn't actually listed on the tooltips at all, it is secretly added into the "roll" used for damage numbers. But it's there and works on weapon-based martial abilities that don't look like they have much power scaling whatsoever (i.e. Knock Down or Force of Anguish). Note that the way the base damage works means that the damage bonus is effectively a multiplicative bonus, making it more impactful than any other damage bonus in the game.
Scrolls: function exactly like spells with one critical difference: instead of using your power level, it substitutes half your Arcana skill for it. Regardless of the spell on the scroll, the spell is treated as PL0. If you have a bonus to your power level (from items, potions, or whatnot), those also boost scroll power 1:1. Might, intellect, and perception have no effect on scrolls. (Note: I suspect it uses half your Arcana skill because scroll strength is already tied to arcana based on the minimum required to use one, so Obsidian didn't want you to get further insane scaling from a scroll of maelstrom, for example.)
Potions/drugs: are influenced by your Alchemy skill. Like scrolls, if you have any bonus to your power level (from items, potions, or whatnot), those also boost potion/drug power level. Nalpazca monks effectively have +10 PL for drugs, which generally means +50% duration with drugs. Pre-1.2, all potions/drugs got their effect boosted by your alchemy skill, but this scaling has been removed with 1.2. (So no more broken uses of Potion of Impediment.) Might, intellect, and perception have no effect on potions.
Explosives: are influenced by your Explosives skill. Bombs used to have a hidden native power level, but as of 1.2 they all have PL0 and have been rebalanced so that they scale appropriately from there. Like with other consumables, if you have any bonus to your power level (from items, potions, etc.) they also boost your explosives skill 1:1. Might, intellect, and perception have no effect on explosives.
Reverse pickpocketing: from stealth, you can reverse pickpocket an explosive (can't be in your quick item slot). It gets a special version of "empower" when it detonates, which grants it +100 accuracy (all but guaranteeing a crit), and +100% to the base damage (multiplicative with other damage bonuses).'
Traps: are weird~! They do their own thing when it comes to PL scaling. First, each point of Mechanics gives a +3 accuracy bonus to traps when they trigger. However, for each of your character levels, it gets a bonus PL, except this PL scaling does not affect damage, they only affect penetration, accuracy, and duration (this may be related to an issue where spells that create "hazard" effects do not do correct damage that scales with stats/abilities). Interestingly, trap duration is not affected by intellect, but trap accuracy is affected by perception. So a level 12 character with 10 mechanics would get a +30 accuracy bonus from mechanics, then another +12 accuracy from PL scaling (listed as an "ability level" bonus in the combat log). Then, if it's a trap with a duration, it gets +60% to duration, and if it's a trap with a penetration value, it gets +3.25 (rounds to 3.3 in display). Anyhoo, this effectively means that for people who keep using traps late game, they will primarily be useful for their debuffs (sorry, caltrops trap).
Monastic Unarmed Training: is also weird! I did my research in a separate thread dedicated to fists.
Additions welcome! I'll edit this post and add in corrections or extra details.
Edited by thelee, 24 October 2018 - 08:22 PM.