The stack is the game zone where spells and abilities are put when they are played and where they wait to resolve. Spells and abilities are put on top of the stack as the first step in being played, and are removed from it as the last step of resolving.
Any spell or ability that uses the stack can be 'responded to' by all players, meaning players have a chance to play spells and abilities with it still on the stack. Since the stack resolves in order from top to bottom, those spells and abilities will resolve before the spell they were played "in response" to. Spells and abilities on the stack resolve one at a time, with a chance for each player to play spells and abilities in between each resolution. Actions that do not use the stack, such as paying costs, playing mana abilities or turning a face-down creature with morph face-up, cannot be responded to.
From the Comprehensive Rules (Innistrad)
- 405. Stack
- 405.1. When a spell is cast, the physical card is put on the stack (see rule 601.2a). When an ability is activated or triggers, it goes on top of the stack without any card associated with it (see rules 602.2a and 603.3).
- 405.2. The stack keeps track of the order that spells and/or abilities were added to it. Each time an object is put on the stack, it's put on top of all objects already there.
- 405.3. If an effect puts two or more objects on the stack at the same time, those controlled by the active player are put on lowest, followed by each other player's objects in APNAP order (see rule 101.4). If a player controls more than one of these objects, that player chooses their relative order on the stack.
- 405.4. Each spell has all the characteristics of the card associated with it. Each activated or triggered ability that's on the stack has the text of the ability that created it and no other characteristics. The controller of a spell is the person who cast it. The controller of an activated ability is the player who activated it. The controller of a triggered ability is the player who controlled the ability's source when it triggered, unless it's a delayed triggered ability. To determine the controller of a delayed triggered, see rules 603.7d-f.
- 405.5. When all players pass in succession, the top (last-added) spell or ability on the stack resolves. If the stack is empty when all players pass, the current step or phase ends and the next begins.
- 405.6. Some things that happen during the game don’t use the stack.
- 405.6a Effects don't go on the stack; they're the result of spells and abilities resolving. Effects may create delayed triggered abilities, however, and these may go on the stack when they trigger (see rule 603.7).
- 405.6b Static abilities continuously generate effects and don't go on the stack. (See rule 604, "Handling Static Abilities.") This includes characteristic-defining abilities such as "[This object] is red" (see rule 604.3).
- 405.6c Mana abilities resolve immediately. If a mana ability produces both mana and another effect, the mana is produced and the other effect happens immediately. If a player had priority before a mana ability was activated, that player gets priority after it resolves. (See rule 605, "Mana Abilities.")
- 405.6d Special actions don't use the stack; they happen immediately. See rule 115, "Special Actions."
- 405.6e Turn-based actions don't use the stack; they happen automatically when certain steps or phases begin. They're dealt with before a player would receive priority (see rule 116.3a). Turn-based actions also happen automatically when each step and phase ends; no player receives priority afterward. See rule 703.
- 405.6f State-based actions don't use the stack; they happen automatically when certain conditions are met. See rule 704. They are dealt with before a player would receive priority. See rule 116.5.
- 405.6g A player may concede the game at any time. That player leaves the game immediately. See rule 104.3a.
- 405.6h If a player leaves a multiplayer game, objects may leave the game, cease to exist, change control, or be exiled as a result. These actions happen immediately. See rule 800.4a.
Actions that use the stack
- casting spells
- activating non-mana abilities
Actions that don't
- playing a land;
- tapping a permanent for mana;
- unmorphing a creature;
- phasing in and out at the start of the untap step (see rule 302.1);
- untapping at the start of the untap step (see rule 302.2).
- drawing a card at the start of the draw step (see rule 304.1);
- declaring attackers at the start of the declare attackers step (see rule 308.1);
- declaring blockers at the start of the declare blockers step (see rule 309.1);
- the active player discarding down to his or her maximum hand size at the start of the cleanup step (see rule 314);
- removing damage from permanents and ending "until end of turn" and "this turn" effects at the start of the cleanup step (see rule 314);
- removing a card with suspend from the game using its suspend ability. 
An action that does not use the stack is called a "game action".