Achievement Points are, simply, points that are gained by completing Achievements.
Each successful completion of an Achievement adds to your total amount of Achievements Points. The amount added is dependent on the achievement completed, see individual achievement articles for point values.
Unlike other types of points (Ability Points or Skill Points) - you do not truly "spend" Achievement Points. Instead, your total number of Achievement points is used to determine what you are able to purchase from the related vendor, Doctor Caligari. The items he has for sale require a minimum amount of Achievement Points in order for you to be allowed to purchase those items. However, the items themselves do not "cost" achievement points, and your number of points do not decrease upon purchasing an item.