Game Mechanic - Priority

Here's a much-requested article - Priority. Here I will be explaining what Priority is, who has it (and when), and what it allows you to use. Examples will also be given, to make this clearer.

Priority is, in essence, a player's right to activate an effect as Chain Link 1 whenever they have priority. As a rule, the Turn Player has Priority when they have successfully summoned a monster, or at the beginning of a Phase or Step. The Opponent has Priority whenever the turn player sets a card, or at the end of a Phase or Step.

Example 1: I Normal Summon The Agent of Creation - Venus. After we agree the summon is successful, I retain priority to activate her effect, by paying 500 Life Points. Since the last action that happened was Venus's successful summon, you are still able to activate your Bottomless Trap Hole, which is Chain Link 2 following Venus's effect. In this case, Bottomless resolves, banishing by Venus, and then Venus's effect resolves, Special Summoning a Mystical Shine Ball from my hand or deck.

Example 2: I enter my Standby Phase, after drawing what is the 4th card in my hand. As a phase of my turn has just begun, I retain my priority to activate Mystical Space Typhoon, from my hand, targeting my opponent's set Spell or Trap. It was Trap Dustshoot - Score! Since I need to have 4 or more cards in my hand for Dustshoot to be activated, he has no time during which Dustshoot can be activated.

Example 3: During my Battle Phase, I declare an attack with my Magical Android (2400) on your Judgment Dragon (3000). You declare no response to the attack. At the beginning of the Damage Step (substep 3, to be precise, but that hardly matters here), I retain priority to activate the effect of Honest, which would increase Magical Android's attack to 5400. As there can only be one Chain during this substep, you still have the option to chain your own Honest, increasing Judgment Dragon's attack to 5400... But it would chain to my Honest, meaning when my Honest resolves I would gain that lovely 5400 attack on my Android, still destroying Judgment Dragon and dealing 2400 battle damage. You, good sir or madam, just wasted an Honest.

Example 4: During my Main Phase, I normal summon Rescue Rabbit. We agree the summon is successful, so I activate Rescue Rabbit's effect, banishing itself as a cost. At no time here are you able to target Rescue Rabbit with Book of Moon, Compulsory Evacuation Device, or Effect Veiler, as he leaves the field before responses can be made to the successful summon.

The common misconception about Priority is that a monster has it in response to its successful summon. The next time somebody claims that a monster has priority, or that they "activate" Priority, I will scream at them. It's tantamount to claiming Bottomless Trap Hole negates a summon.

I'm renowned for my long posts, so here's a quick summary:
The turn player has priority in response to every action they perform that does not start a chain. Setting a card, starting a phase or step, declaring an attack, summoning a monster, or a chain resolving. If you do not use your priority at any of these times, you pass priority to your opponent, allowing them to start a chain. You also pass priority to your opponent at the end of each phase. Priority does not allow you to resolve an effect first, it merely allows you to start a chain. No part of priority stops your opponent responding to an action.