Priority

Priority can be defined as "A player's 'right' to be able to activate his/her effect, ensuring its spot on Chain Link 1. Priority does not allow players to automatically activate and resolve effects without a chain; it simply assures a card's activation (not resolution), and guarantees a place on Chain Link 1. How the chain resolves is up to the players.

Priority tells you who can activate an effect at any certain point in the game. Note that cards never have priority'; it is the players who have priority. The Turn Player always starts with priority.

Example 1
Player A has a face-up "D.D. Warrior Lady". He Sets a "Bottomless Trap Hole" and ends his turn.

Player B Special Summons "Chaos Sorcerer".

Player B, now being the turn player, says he wishes to retain his priority to activate the effect of "Chaos Sorcerer". "Chaos Sorcerer"'s Ignition Effect then activates, targeting "D.D. Warrior Lady". "Chaos Sorcerer"'s effect is now Chain Link 1. Player A then activates his "Bottomless Trap Hole", since the Special Summon of "Chaos Sorcerer" was the last action to have resolved on the field. "Bottomless Trap Hole" is now Chain Link 2.

The chain now resolves in reverse:
 * Chain Link 2: "Bottomless Trap Hole" resolves. "Chaos Sorcerer" is destroyed, and removed from play.
 * Chain Link 1: "Chaos Sorcerer's" effect resolves. "D.D. Warrior Lady" is now removed from play.

Always remember that Monster Effects, unless they specify otherwise, can have their effects resolved without being present on the field.

Example 2
Now take the same scenario, but instead of "Bottomless Trap Hole", we can use "Book of Moon" as the example.

The resolution of this chain works as follows:
 * Chain Link 2: "Book of Moon", targeting "D.D. Warrior Lady" resolves, flipping "D.D. Warrior Lady" to face-down Defense Position.
 * Chain Link 1: "Chaos Sorcerer"'s Ignition Effect tries to resolve, but since his target is now face-down, his effect disappears, because "Chaos Sorcerer" can only remove face-up monsters from play.

Example 3
Yet another example of priority can be seen in this example:

Player A ends his turn with a face-up "Mystical Elf" and a set "Trap Hole".

Player B Normal Summons "Exiled Force". Since "Exiled Force" has an Ignition Effect, Player B can use 'priority to activate its effect, which becomes Chain Link 1. However, the cost of "Exiled Force"'s effect is to Tribute itself. Since costs are mandatory to activate a card (not resolve the effect), the cost of "Exiled Force" is immediately carried out. "Exiled Force" is removed from the field, and Player A's "Trap Hole" will no longer have a monster to resolve its effect for.

Passing Priority
One can pass priority by simply stating "I pass priority".

Optional Trigger Monster Effects, like "Mobius the Frost Monarch", are not Ignition Effects, so their effect(s) can only be activated when the condition that triggers them was the last action to have resolved on the field.

Priority will automatically be passed to the opponent if the Summoned monster has a Compulsory Monster Effect, like "Zaborg the Thunder Monarch".

Priority is also automatically passed when the Turn Player declares an end to any Phase.

Exceptions
Turn Priority normally cannot be ignored. However, if the opponent has a card that must be activated at a specific time, such as the Standby or Main Phase of the opponent's turn, these cards can be activated even if the Turn Player has no cards to play at that time. If the card is played during the end of the Main Phase the opponent has passed up their opportunity to play their card first. However, a replay can be demanded for any other situation if the Turn Player wishes to play a Chain Link 1. To avoid this dispute, it is etiquette to ask the Turn Player if they would like to play a card during a respective phase (as declining to play a card results in passing their Priority to the Opponent).