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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.
Dark Power is affected by the light level of the province in question. Regular daylight is counted as zero Darkness. Caves and Deep Seas have 3 Darkness. A few spells can create unusual Darkness levels, even negative values. Whatever it is, multiply this by the value of the trait.
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.
Storm Power is binary; it 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. The reason is similar to Dark Power's interaction with Darkvision, except that the penalty doesn't interact much with the bonus; Storms halve Precision and prevent Flying, for those without Storm Immunity.
Power of the Turning Year is also binary, providing a bonus only on 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.
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.