A Power trait governs the four unit attributes that are understood as physical power in real life: Strength, utilization of strength (Attack Skill), agility (Defence Skill), and speed (Combat Speed). There are many of these, but only seven are used in-game:
Fire Power, Cold Power, Chaos Power, and Magic Power are very "swingy"; this is because they work based on the local Scales. You take the trait's value, and you multiply it by the point value of the relevant scale; that modifies the stats governed by the trait. A negative point value results in a stat penalty instead of a bonus, of course.
Chaos Power is special: it also counts Unrest. For every 15 points of local Unrest, add 1 to the Turmoil value of the local province, but only for the sake of this trait. In any case, Turmoil caps at 5, regardless of how high Unrest gets.
Magic Power also affects Magic Resistance, negating the positive and negative effects of Magic/Drain scales on the stat. It also affects Stealth; only, instead of +1 per positive point, it gives +10.
Philosopher is not a Power Trait; it works in the exact same way, but it only affects Research Skill. It scales positively with Sloth, and negatively with Productivity, because Philosophers are lazy.
Two of the power traits never cause a stat penalty; they either apply their listed point value to the combat stats, or they don't.
Storm Power only cares if a Storm or a Blizzard is in the current battle. If there's none, there's zero Storm Power; otherwise, the bonus is 1. Whatever it is, multiply this by the value of the trait.
Storm Power typically comes with Storm Immunity. For those without Storm Immunity, Storms halve Precision and prevent Flying. Storms also cause half of nonmagical missiles to end up nowhere, which may or may not affect the Storm-Immune.
Power of the Turning Year is also binary, providing a bonus only in Midspring, Midsummer, Midfall, and Midwinter. Turn 1 is Midspring, Turn 4 is Midsummer, Turn 7 is Midfall, and Turn 10 is Midwinter. A year is 12 turns; you can extrapolate the rest. The game does not actually use this terminology, instead referring to each as "the height of their season"; in the user interface, these turns are merely referred to as Spring, Summer, Autumn, and Winter.
Power of the Turning Year does not affect Combat Speed, for some reason.
There's also a similar series of traits, "Strength of the Season", which take effect in certain seasons; these do not interact with any of the Power attributes, however, but rather Hit Points.
Dark Power is a very weird power trait; in most cases it's a binary trait, but magic can make it scalar.
The binary element of Darkness cares about whether or not the fight takes place in one of these conditions. In this case, the "Darkness value" is 3.
Several spells can force this standard darkness in battle, but there are also magical effects that force "odd" values. Darkness and Utterdark make it 6, for example, while Fire Storm reduces it by 1 (possibly to a negative value) and a local Eternal Pyre sets it to zero. Where Darkness is zero, Dark Skies makes it 1 (if Eternal Pyre isn't responsible), while Solar Brilliance makes it negative 1.
Dark Power typically comes with Darkvision. This is because Darkness is quite severe for everyone who doesn't have it (or at least a superior Sight Trait). A point of Darkness reduces Attack Skill, Defence Skill, and Precision for everyone who doesn't have 100% Darkvision; by 0.5 if they have 50% Darkvision (which then gets rounded down), or by 1 if they don't have any Darkvision at all. Negative points of Darkness don't help anyone.