This article is from the Magic: The Gathering Rules FAQ, by Patrik Linell (pls@claymore.nu) with numerous contributions by others.
The Stack is a timing system. It's a zone of play, like your
graveyard or your library. When a spell is announced, it cannot take its
effect straight away. Instead, it's put on top of the stack, and other
players are given the opportunity to respond to it with their own spells.
The responses will resolve first, and when people have finished responding,
eventually the original spell will resolve. (Note that this also applies to
abilities.) The player who "has priority" at any particular time is the one
who's currently allowed to add things to the stack. Here's a brief
flowchart showing how the system works.
In each step of the turn, the active player has priority to begin with.
I: The player who has priority may announce a spell: he puts the spell
onto the stack, chooses its mode and its targets, and pays its costs. If he
chooses to do so, then he gets priority again- repeat this step. Otherwise,
he "passes priority" to his opponent- go to [II].
II: Now, that player's opponent has priority, so he can play his own
spells on the top of the stack. If he chooses to do so, then he'll be the
one who priority again- we return to [I]. Otherwise, we go on to [III].
III: The players have both passed, so they've both declared that they're
not going to respond to the current spell. If the stack is empty, then the
current step of this turn will end. Otherwise, the top item of the stack
resolves and is removed (in the case of an Instant or Sorcery it will go to
the graveyard. Other types of spell will go into play, and pseudospells
from abilities will be removed from the game). After that, the active
player gets priority again. We return to I.
 
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