From MTG Salvation Wiki
An activated ability is an ability that can be played by a player by paying a cost. The cost can be almost anything, such as sacrificing a permanent, paying mana, paying life or tapping the permanent with the ability (
). Activated abilities are usually found on permanents, but can sometimes be used on cards in other zones, such as the graveyard or a player's hand.
Only an object's controller (or owner, if it has no controller) can play its activated abilities unless the ability says otherwise. Also, you can usually play an activated ability any time you have priority, unless it says otherwise. Excpet for those that are mana abilities, they use the stack.
A player cannot play an activated ability of a creature that includes the tap symbol () in its cost unless he or she has controlled the creature continuously since the start of his or her most recent turn. This condition is informally known as "summoning sickness."
Activated abilities cannot be countered by spells or abilities that counter spells, because they aren't spells. However, there are cards, such as Stifle, Squelch and Voidslime, that can be used to counter these abilities.
[edit] Examples
Below are examples of different activated abilities.
Lore Broker's activated ability can be activated any time the controlling player could play an instant, which means he or she must have priority to play it. The cost includes the tap symbol , which means that because Lore Broker is a creature, the ability cannot be played if the controller has not controlled it from the beginning of his or her turn. (This is informally known as "Summoning Sickness".)
Nezumi Bone-Reader's activated ability's cost does not include the symbol, which means it can be used even if it has summoning sickness. It can however only be used in the controller's main phases when he or she has priority and the stack is empty.
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Orochi Leafcaller - 
Creature — Snake Shaman
1/1
: Add one mana of any color to your mana pool.
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Because Orochi Leafcaller's activated ability produces mana, it is a mana ability, and thus it doesn't use the stack. This means the controller doesn't lose priority when he or she plays this ability, and that in turn means that an opponent cannot respond to this activation.
Squallmonger's ability can be activated by other players than the controller, because the card explicitly says so. It still requires a player to have priority though.
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Nullmage Shepherd - 
Creature — Elf Shaman
2/4
Tap four untapped creatures you control: Destroy target artifact or enchantment.
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Because Nullmage Shephard's ability's cost doesn't include the symbol, the controller is allowed to tap the Nullmage Shephard, or other creatures, with summoning sickness.
Ghost-Lit Raider has two activated abilities. The card doesn't explicitly say when you can play the two different abilities, but it is inferred by the cost.
The first requires the controller to tap it, an thus requires the card to be in play, as only permanents can be tapped. The second, which incidentally has the ability word "Channel", requires the controller to discard it from his or her hand. This implies that it can only be played when it is in its controllers hand.
Shard Phoenix also has two activated abilities. The first requires Shard Phoenix to be in play, and for its controller to have priority. The second ability can only be played when the card is in its owner's graveyard and only during its owners upkeep step.
Sekki, Seasons' Guide has one activated ability which can only be played when the card is in a graveyard.
[edit] Formal rules
From the Comprehensive Rules: (February 1, 2009)
- 403. Activated Abilities
- 403.1. An activated ability is written as “[cost]: [effect].” The activation cost is everything before the colon (:). An ability’s activation cost must be paid by the player who is playing it.
- 403.2. Only an object’s controller (or its owner, if it doesn’t have a controller) can play its activated ability unless the object specifically says otherwise.
- 403.3. If an activated ability has a restriction on its use (for example, “Play this ability only once each turn”), the restriction continues to apply to that object even if its controller changes.
- 403.4. A creature's activated ability with the tap symbol (}) or the untap symbol (
 ) in its activation cost can't be played unless the creature has been under its controller's control since the start of his or her most recent turn. Ignore this rule for creatures with haste (see Rule 502.5).
- '403.4a' If an object acquires an activated ability with a restriction on its use from another object, that restriction applies only to that ability as acquired from that object. It doesn’t apply to other, identically worded abilities.
- 403.5. Activated abilities that read “Play this ability only any time you could play a sorcery” mean the player must follow the timing rules for playing a sorcery, though the ability isn’t actually a sorcery. Activated abilities that read “Play this ability only any time you could play an instant” mean the player must follow the timing rules for playing an instant, though the ability isn’t actually an instant.
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For your reference [CR 409]
- 409. Playing Spells and Activated Abilities
- 409.1. Playing a spell or activated ability follows the steps listed below, in order. If, at any point during the playing of a spell or ability, a player is unable to comply with any of the steps listed below, the spell was played illegally; the game returns to the moment before that spell or ability was played (see rule 422, “Handling Illegal Actions”). Announcements and payments can’t be altered after they’ve been made.
- 409.1a The player announces that he or she is playing the spell or activated ability. It moves from the zone it’s in to the stack and remains there until it’s countered or resolves. In the case of spells, the physical card goes onto the stack. In the case of activated abilities, the ability goes onto the stack without any card associated with it. If the ability is being played from a hidden zone, the card with that ability is revealed. Each spell has all the characteristics of the card associated with it. Each activated ability on the stack has the text of the ability that created it, and no other characteristics. The controller of a spell is the player who played the spell. The controller of an activated ability is the player who played the ability.
- 409.1b If the spell or ability is modal (uses the phrase “Choose one —” or “[specified player] chooses one —”), the player announces the mode choice. If the player wishes to splice any cards onto the spell, he or she reveals those cards in his or her hand. If the spell or ability has a variable mana cost (indicated by {X}) or some other variable cost, the player announces the value of that variable at this time. If the spell or ability has alternative, additional, or other special costs (such as buyback, kicker, or convoke costs), the player announces his or her intentions to pay any or all of those costs (see rule 409.1f). You can’t apply two alternative methods of playing or two alternative costs to a single spell or ability. Previously made choices (such as choosing to play a spell with flashback from his or her graveyard or choosing to play a creature with morph face down) may restrict the player’s options when making these choices.
- 409.1c If the spell or ability requires any targets, the player first announces how many targets he or she will choose (if the spell or ability has a variable number of targets), then announces the targets themselves. A player can’t play a spell or ability unless he or she chooses the required number of legal targets. The same target can’t be chosen multiple times for any one instance of the word “target” on the spell or ability. If the spell or ability uses the word “target” in multiple places, the same object, player, or zone can be chosen once for each instance of the word “target” (as long as it fits the targeting criteria).
- Example
If an ability reads “Tap two target creatures,” then the same target can’t be chosen twice; the ability requires two different legal targets. An ability that reads “Destroy target artifact and target land,” however, can target the same artifact land twice because it uses the word “target” in multiple places.
- 409.1d If the spell or ability targets one or more targets only if an alternative, additional, or special cost (such as a buyback or kicker cost) is paid for it, or if a particular mode is chosen for it, its controller chooses those targets only if he or she announced the intention to pay that cost or chose that mode. Otherwise, the spell or ability is played as though it did not have those targets.
- 409.1e If the spell or ability affects several targets in different ways, the player announces how it will affect each target. If the spell or ability requires the player to divide or distribute an effect (such as damage or counters) among one or more targets, or any number of untargeted objects or players, the player announces the division. Each of these targets, objects, or players must receive at least one of whatever is being divided.
- 409.1f The player determines the total cost of the spell or ability. Usually this is just the mana cost (for spells) or activation cost (for abilities). Some cards list additional or alternative costs in their text, and some effects may increase or reduce the cost to pay. Costs may include paying mana, tapping permanents, sacrificing permanents, discarding cards, and so on. The total cost is the mana cost, activation cost, or alternative cost, plus all cost increases and minus all cost reductions. Once the total cost is determined, it becomes “locked in.” If effects would change the total cost after this time, they have no effect.
- 409.1g If the total cost includes a mana payment, the player then has a chance to play mana abilities (see rule 411, “Playing Mana Abilities”). Mana abilities must be played before costs are paid.
- 409.1h The player pays the total cost in any order. Partial payments are not allowed.
- Example
You play Death Bomb, which costs  and has an additional cost of sacrificing a creature. You sacrifice Thunderscape Familiar, whose effect makes your black spells cost  less to play. Because a spell’s total cost is “locked in” before payments are actually made, you pay  , not , even though you’re sacrificing the Familiar.
- 409.1i Once the steps described in 409.1a–409.1h are completed, the spell or ability becomes played. Any abilities that trigger on a spell or ability being played or put onto the stack trigger at this time. If the spell or ability’s controller had priority before playing it, he or she gets priority.
- 409.2. Some spells and abilities specify that one of their controller’s opponents does something the controller would normally do while it’s being played, such as choose a mode, choose targets, or choose how the spell or ability will affect its targets. In these cases, the opponent does so when the spell or ability’s controller normally would do so.
- 409.2a If there is more than one opponent who could make such a choice, the spell or ability’s controller decides which of those opponents will make the choice.
- 409.2b If the spell or ability instructs its controller and another player to do something at the same time as the spell or ability is being played, the spell’s controller goes first, then the other player. This is an exception to rule 103.4.
- 409.3. Playing a spell or ability that alters costs won’t do anything to spells and abilities that are already on the stack.
- 409.4. A player can’t begin to play a spell or activated ability that’s prohibited from being played by an effect.
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). Activated abilities are usually found on permanents, but can sometimes be used on cards in other zones, such as the graveyard or a player's hand.
