Triggered abilities, from condition to chain

A triggered ability is an event, a check, and a place on the chain. The event happens, the game checks the condition, and the ability goes on the chain as an item of its own that opponents can answer. Most rulings are about when that check happens and what the ability still knows afterwards.

Worked example. You play Lecturing Yordle, "When you play me, draw 1". The Yordle is a unit, so it resolves immediately and enters the board (rule 337.2). Its Play Effect is then put on the chain as a Pending Item (rule 383.4.a.2), where the opponent can respond. Say they kill the Yordle in response. The draw is already its own item on the chain, a triggered ability behaves like an activated ability once it is there (rule 383.3), so it still resolves. Compare Diana, No Longer Human, "When you play a spell, give me +2 Might this turn": it triggers on another card being played, so it is not a Play Effect (rule 383.4.a.4), and if the spell is countered it never triggers (rule 420.4.a and rule 420.4.a.1).

Recognizing one. A triggered ability usually has "when" followed by an event, "at" followed by a point in the turn, or "the Nth time" (rule 383.1). It has a Condition and an Effect (rule 383.2). A conditional statement immediately after the Condition is part of the Condition and must be true for it to be fulfilled, but once the ability is on the chain it still resolves if that statement stops being true (rule 383.2.a.1).

When the condition is checked. The Condition is evaluated after the inciting event has been processed (rule 383.2.c). An object that enters the zone its trigger works from at that same moment can still trigger (rule 383.2.c.1), and one that leaves that zone at that moment cannot (rule 383.2.c.2). The exception is a trigger whose condition is its own source leaving the board, such as Deathknell or Treasure Trove: it is processed just before the object changes zones, after replacement effects, with information about the object noted for later (rule 383.2.c.3, rule 383.2.c.3.a and rule 383.2.c.3.b).

Choices and costs. A triggered ability is placed on the chain in Open or Closed States, on any player's turn (rule 383.3.c). If its Effect starts with "you may", the choice is made during finalization, and choosing not to removes it from the chain and counts it as not triggered (rule 383.3.a and rule 383.3.a.2). A "you may" later in the Effect is decided on resolution (rule 383.3.a.3). A cost at the beginning of the Effect, or right after that first "you may", is the ability's base cost and must be paid to finalize it (rule 383.3.b and rule 383.3.b.1). A cost further into the effect is paid on resolution (rule 204.3.a). An ability that triggers "once each turn" only triggers that many times (rule 383.3.e.1).

Ordering. If several triggered abilities trigger at once, their controller chooses the order to put them on the chain (rule 383.3.d). If different players control them, the Turn Player orders theirs first, then each player in Turn Order (rule 383.3.d.1). The newest item resolves first (rule 340.1), so the last player in that order sees their abilities resolve first.

Named triggers. Conquer and Hold Effects still trigger if the point from conquering or holding is negated or replaced (rule 383.4.c.2.c and rule 383.4.d.2.c). Attack and Defend triggers fire when a unit first gains the designation in a combat and have their condition checked only once per combat (rule 383.4.e, rule 383.4.e.2.a and rule 383.4.f.2.a). In tournaments the accountable player must acknowledge a trigger by the time it would have an observable impact, or it is forgotten (TR 506.1, TR 506.3 and TR 506.4).

Common mistake. "I responded to the unit, so I stopped its trigger." The trigger is a separate item and still resolves.