The following page of the guide to Disgaea RPG answers the question what are action points (AP) and what they are used for.
The purple bar with the current amount of APs is always visible in the upper left corner of the screen. These points are necessary to perform the main missions and go through the levels in the Dark Gates.
Action points renew at midnight every day. However, there are ways to earn extra points, which can even go beyond the scale.
AP potions renew the Action Points, obviously. To use them, click the plus sign next to the AP bar. Potions can be purchased at the store, received as a reward when completing missions, or as gifts for logging into the game.
Every 8 hours, we can use the Doping Roulette option in the hospital, and win up to 200 AP!