Trigger
In Noita, trigger (or with trigger) refers generally to spells that create projectiles that can be given a payload, consisting of one or more spells, that are released at the location of the trigger projectile under specific triggering conditions.
There are three types of trigger, with different triggering conditions. A spell 'with Trigger' specifically refers to those that release the payload when they hit a target, the others, Timer and Expiration have more specific names, but all three kinds can be referred to as triggers.
Payload
Triggering Conditions
Projectiles can have (or be given) a payload of spells, to be released...
Payload Selection
The spells that end up inside the trigger payload are determined in the same way that the wand determines which spells end up in each shot, using a draw counter and drawing spells until the counter gets to zero. Most trigger spells (and those created using Add Trigger/Add Timer/Add Expiration Trigger) start creating a payload with a draw count of 1. Spark Bolt with Double Trigger and Delayed Spellcast start with 2, and 3, respectively.
The spells after the trigger spell are drawn in sequence, and if they add to the draw counter then it is kept from hitting zero and more spells end up in the payload. Wrapping can happen inside the payload scope, with the wrapped-into spells ending up inside the payload too. This can be a very useful trick, especially if used with Wand Refresh so that you end up with a second copy of the whole wand inside the payload created by the first.
Payload Isolation
The trigger payload is isolated from the effects and modifications applied by spells outside of it. For example, any cast delay applied by spells is ignored if they are inside the payload, and any damage increasing modifiers applied to the payload do not affect the trigger projectile or other projectiles it is multicast with.
The speed reduction applied by Heavy Shot only affects the Spitter Bolts inside the payload - speed modifications are contained. However, so are damage modifications - but that isn't necessarily a bad thing. The Spark Bolt doesn't gain any damage from Heavy Shot, but the Spitter Bolts don't lose any damage from Light Shot.

Some properties (e.g. Recharge Time) are not isolated in this way - Recharge Time modifications inside the trigger payload count toward the overall recharge time of the wand.
You can exploit these differences in property scope in wand building. For example, Spread is isolated, so by putting Heavy Spread modifiers into the payload the negative effect of the spread increase is avoided (the Spitter Bolt is going to be spawned right on top of the enemy - its direction doesn't matter), while still getting the positive effect of reduced Recharge Time.

Many tricks in wand building rely on properly isolating modifications so they affect only some spells.
Running out of Mana
When firing a spell with a trigger or timer attribute, Noita checks for the combined mana drain of the trigger spell and the spell after it. If the wand has enough reserves to fire the whole combination, it's all cast as expected: first the trigger spell, then the spell after it when the conditions are met.
If the wand does NOT have enough mana reserves to fire the whole sequence, the trigger spell itself is prioritized and (some or all) spells after it are omitted. An exception is if the trigger spell is too expensive even by itself, in which case cheaper spells after the trigger will cast instead without the trigger (this can potentially be dangerous).
List of spells with triggers
See Also