Activity Points is a sub-category stat and a third-bar used on various classes, the purpose of this stat vary and the way to acquire and use it also change with the classes. Some is to increase damage or to generate bonuses, others may use as a Requisite to cast a skill, and so on.
Some classes may have AP but they wont use it at all, you can check on this page all classes that actually use it.
Activity Points are show on your character information with AP 0/X meaning X is the maximum you can store, different classes have different amount of AP. You can also see your points bellow your SP/HP bars, with a Yellow Color.
Classes
Monks - Sura
Monk Combo
For Monks your Activity Points will be used as Combo Points, and a maximum of 5 can be stored.
While fighting you can see your bar filing up, you can store all the 5 using the monk skills and expend it all at once on a skill that consumes it like Tiger Fist and Throw Spirit Sphere
Each Combo Point will buff those skill damage by a % of damage See skills.
Combo Points can be used anytime, you dont need to wait it fill up at 5.
Combo Points have no Cooldown of any kind, you can acquire points everytime you use skills, meaning the speed you cast those skills is the same you acquire Points.
On Sura you keep with the same system but your maximum points is increased to 10.
Sura also receive more skills to use and buff your Monk Skills.
Gunslinger
Gunslinger Bullets
Gunslinger Activity Points is used in form of a Bullet System. You have only 10 Bullets in all Guns.
Your points from Axe and Mecha are not shared, if you summon your mecha while with Axe points, all your points will be lost, the same apply from Mecha to Axe.