How much damage?

If the source of the damage is a prepared danger, use the danger's stats. For example, if your move is to have a hagr pluck a PC up from the ground and start choking them, then you'd deal the hagr's 1d10+3 damage.

When multiple combatants deal damage at the same time, roll the single highest damage die among them and add +1 for each additional combatant. The tags from all the combatants apply. So if six coedwig (thorn-spirits, dealing d6 damage, 1 piercing) and a birdwraig (tree spirit, d8+3 damage, forceful) all pound on the Heavy at once, the Heavy is taking d8+9 damage (1 piercing, forceful)—and probably having a very bad day.

If the damage isn't associated with a prepared danger, then ask yourself what it could likely do to a normal person. If the worst it's likely to do is…

… cause bruises and scrapes, pain, or light burns, it inflicts d4 damage.
… cause nasty flesh wounds, bruises, or burns, it inflicts d6 damage.
… break bones, cause deep/wide burns, or debilitate with pain, it inflicts d8 damage.
… cause death or dismemberment, it inflicts d10 damage.

Subtract the victim's armor rating from the damage they take. This can reduce the damage to 0, meaning they don't lose any HP. Good for them! They still got hit/ tossed around/whatever, but it hasn't used up any of their vitality, luck, or plot armor.

If the attack has the "[n] piercing" tag, ignore [n] points of armor for that attack. So, if you deal 5 damage (1 piercing) and the target has 2 armor, their armor is effectively only 1 for that attack; they lose 4 HP (but their armor value is still 2 against future attacks).

If the attack "ignores armor," then the armor doesn't apply. They take the full damage. Apply this tag on the fly when the fiction dictates: chainmail doesn't help when you're engulfed in flames, and no amount of physical armor protects you from a psychic assault.

A handful of player-facing moves tell the player to "lose [n] HP" (as opposed to "take [n] damage"). Armor never applies to this sort of HP loss.