Let's go with the basic, ultimate question: "WHAT is priority?"
Priority is the right of the turn player to perform certain actions before the opponent does.
Wait, that's it? That miserable line is the solution to every priority question and the very basic backbone of how a player's turn is developed? Yes, that's it, as simple as it looks.
Now, let's take a second look at that statement. Of course, once you read it, you realize that it's really, really, broad. It's like telling you "go buy stuff sometime". So first things first, let's look at it a little closer.
The main thing you'll notice is that priority is a "right". That is to say, the turn player always possesses priority, and this is a fact. The only times the turn player doesn't have priority is when the player already used it, or when he or she chose not to use it. We'll deal with those situations later. In my opinion, the easiest way to grasp this concept is to think of a tennis match, and to consider that the turn player always gets to serve. If you don't like tennis, like me, then think of the videogame Pong Q_Q. Until the turn player serves, nothing can happen, as the ball will stay still. Similarly, until the turn player takes an action using priority, nothing happens in the Duel.
This is important for places where there is a big lack of knowlege of how a player is meant to behave during a turn. We've all came across unexperienced players, specially younger ones, who simply flip Cards whenever they feel like, claiming that they "activated it before you", or that they chained about 5 Cards to your own. When you think of the boring tennis player that refuses to serve, one can compare it to how the opponent must wait until the ball starts moving before taking actions. And certainly, tennis isn't a matter of speed: A player doesn't get to hit the ball several times before the opponent does. They alternate with each other.
Are you getting the picture of where this is going? This is the very mechanic behind how a chain is built! Take a look at the following chain:
Player A: Activates Heavy Storm
Player B: Chains Imperial Order
Player A: Chains Seven Tools of the Bandit
So Player A "served" by activating Heavy Storm. Player B can now start moving, and he hits back hard with Imperial Order. Player A replies with Seven Tools of the Bandit. Player B doesn't manage to hit this speedy ball back, and so, Player A "scores". Back to YGO, Seven Tools "went through" player B, which didn't respond with another Card that could negate Seven Tools, so Imperial Order's activation is negated, and Heavy Storm destroys a bunch of Spell and Trap Cards.
Certainly, the tennis example doesn't apply to every possible chain, but I find it to be helpful to at least get things going. The mere idea of mixing sports and Yugioh terrifies Fox Sports itself.
Moving on, let's look at the next part. Exactly, which actions can the turn player perform while ruining everyone's fun by keeping the ball? The answer is "all of them". Every action you take during your turn involves using priority. This isn't too clear at first sight, because the few times players remember that priority even exists, is when a monster is Summoned. However, priority applies during the entire turn, even at obscure times like "the beginning of Main Phase 1" (the only possible way to activate Cold Wave), or to manipulate effects during the Standby Phase (if I wish to activate The Immortal Bushi before Future Fusion Summons a monster).
Thanks to the limitations within a given Phase, not every action can be performed at any time. For example, you can't Normal Summon a monster during the End Phase. That really helps narrowing down this list of actions:
1) During the Draw, Standby, Battle, and End Phases, you can't perform a Normal Summon, a Set, manually activate a Spell Speed 1 effect (Normal Spell Card or Ignition Effect, for example). That leaves us with the turn player being able to activate a Spell Speed 2 or higher effect by using priority. For example, during the Battle Phase, you can use priority to activate Magician's Circle when your monster attacks.
There are other effects, which activate at these phases, that can be manipulated with priority, but those are too detailed for this article.
2) During the Main Phases, it gets a little more complicated. The simpliest, most popular scenario is when the last thing to happen is the turn player performing a Summon. At that point, the turn player can activate an Ignition Effect or a Spell Speed 2 or higher effect by using priority. This is the bread and butter of introducing a player to the concept of priority: "Activate Rescue Cat's effect with priority", "Activate Exiled Force with priority". It becomes the prime example of priority, because if there's a time where you want to activate an effect before the opponent does, then that's when you want your newly Summoned monster to change the tides using its effect before your opponent gets rid of it with Bottomless Trap Hole. Do note that this reffers to any Ignition Effect you control, even of a monster that you had previously Summoned before (for example, you can use priority to activate Chaos Sorcerer's effect when you Normal Summon Jinzo).
If the last thing to happen was NOT a Summon, then the turn player only has priority to activate a Spell Speed 2 or higher effect. For example, if the last thing to happen was adding a monster to your Hand via Reinforcement of the Army, you cannot use priority to Normal Summon that monster. Similarly, if the last thing to happen was sending Destiny Hero - Malicious to the Graveyard, you cannot use priority to activate its Ignition Effect.
And the last scenario, possibly the weirdest one at first sight: If you aren't responding to any previous event (a Card's destruction, adding a Card to the Hand, a Summon, a Battle Position change, etc.), then you have priority to perform any action, specially those disallowed in the first item (performing a Normal Summon, change a Battle Position, activate a Normal Spell Card, etc.). For example, if a DARK monster is destroyed, and neither player wishes to respond to this destruction, the turn player is free to perform certain actions. Namely, Special Summoning Dark Armed Dragon. It becomes a common question whether one can jump the gun and activate D.D. Crow or similar once the opponent is verifying the Graveyard throughly to check if there are exactly 3 DARK monsters. But since you didn't wish to respond to the addition of the third DARK monster to the Graveyard, the turn player now has priority to Summon DAD.
Stay tuned for part 2, where I'll address the rest of the definition of priority, and the cases in which the turn player passes it. If you have any questions, feel free to drop me an e-mail at ness00[at]gmail[dot]com.