Action Points

Action Points refers to one of two concepts in tabletop games.

The first is a system where every action a character does on his turn costs action points, but the character can do as many actions as he has the points for. For example, drawing a weapon, moving and firing may all require action points. This is quite difficult to balance, since it is quite hard to make abilities more useful than attacking more often. This is much rarer in tabletop games than video games because it is hard to keep track of in non-electronic formats.

The second and more common is a system of limited resources that allow players to swing the narrative in their favor. Spending an action point might allow a character to reroll a failed roll, make their ability work better or conveniently have a useful item. Since these are most often used out of turn or creatively they are quite rare in video games.

Dungeons & Dragons

While Dungeons & Dragons has never included action points as a core rule, the 3rd edition version of Unearthed Arcana and Eberron both include a variant of the system as optional rules.

Pathfinder introduced another optional variant called "Hero Points" in the Advanced Player's Guide.

Mutants and Masterminds

Mutants and Masterminds call these "Hero Points". At the start of a session, each player gets one point but they lose all hero points when the session ends. Everytime a hero is inconvenienced by a "Complication" they selected at character creation, such as being forced to do something by their personality, a loved one is put in danger, or their special weakness is used against them, they are award a hero point. Hero points can be used to reroll die with the new roll having a minimum value of 11, temporarily gain advantages (You might reveal you can speak a language or have a friend that can provide a needed service), add useful details to a scene (Why yes, there is a fire hydrant near where you're fighting the fire elemental.), have a piece of equipment on hand, use an ability outside of your turn, or to cure conditions. You can cure conditions at the same time you acquire them, which effectively allows you to use extra effort (make an ability more powerful at the cost of exhausting yourself) and power stunt (use your powers in an unusual way).

Saga Edition

Star Wars Roleplaying Game Saga Edition call these Force Points. These represent destiny tugging a bit to favor you even if you aren't able to channel the force (even droids get them). These can also be expended to increase die rolls by 1d6, but since death is permanent in Star Wars, they're best saved for increasing defenses or on shrugging off the darkside. Each time a character levels up their Force Point total is set to 5 + (level/2), losing unspent points. Some abilities can be enhanced by or (less usefully) require spending Force Points.

Jedi Knights get access to the ability Force Point Recovery, which effectively gives them a free Force Point per encounter. The Clone Wars book has a feat that gives you a temporary Force Point one per encounter if an ally hits you with a force power. While on it's own this would be nifty if you were allied with a buffing force user, in the d20 system you are your own ally so it's one of the best feats in the game since you can trigger it yourself. These two abilities make abilities that require force points actually useful. A few other ways of getting temporary force points exist, but they aren't as reliable.

Force Points have a big brother in Destiny Points. These are more powerful allowing you to automatically score a critical hit or force an attack to miss among other options. You only get one each level, though they are cumulative. You lose them when you fulfill your destiny, though they may be given to other PCs when you do.