Activated abilities are abilities that you choose to activate. They are written in the form of "Cost: Effect" where the cost can be energy, exhausting the card, or something else entirely.
[[Animated Sword]]
For example, Animated Sword has the text: "2, Exhaust: Target minion you control gets +3/+1 until the next clear." This is an activated ability that costs 2 and exhausts the permanent, then gives a minion you control +3/+1 until the next clear phase.
By default, activated abilities are not Nimble, meaning they can only be used when you could invoke a minion. Non-Nimble abilities can only be used during your main phase, when the stack is empty, and you have priority. However, some activated abilities are Nimble and can be used at any time you have priority. Also note that the card must be in play and you must be able to pay its cost to use its activated ability.
There are some exceptions to this rule, such as abilities that say "This ability is Nimble." Those can be used at any time you have priority.
[[Hungry Ghoul]]
Hungry Ghoul's ability is an example of a Nimble ability. It can be activated at any time, meaning you can liquidate one of your own minions in response to removal, for example.
[[Stitched Ghoul]]
Stitched Ghoul is one of the few examples of a card whose abilities can be activated from a zone other than the field, in this case, from your hand. You know this because it says "Discard this card" as part of the cost, and cards can only be discarded from your hand. Also note its first ability is Nimble.
You can use an activated ability by right clicking on the card, then selecting the ability from the menu that appears. If you have the necessary resources to pay the cost, the ability will be activated. If you cannot pay the ability's cost, then it will not show up in the menu.