In Action Point (AP) Allowance System games, each player is allotted a certain amount of points per round. These points can be spent on available actions, until the player does not have enough remaining to "purchase" any more actions. This method grants the player greater freedom over how to execute his or her options. Pandemic is an example of a game that uses this mechanic. In Pandemic, players are given 4 action points to be allocated between several actions: Movement, Air Travel, Special Action, and Special Ability.
The earliest example of a game listed on Boardgamegeek that uses AP's is Special Train (1948).