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 and a character might cut movement short to be able to afford a second attack. 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. The basis is that at creation, each character has an extremely limited pool of these points, usable for one of a few remarkable effects, like adding to poor rolls, preventing death, and recharging expendable resources like spells. What offsets this feature from being OP as hell, however, is that coming across new Action Points is incredibly rare - rare enough that you only get new ones every level - and that they have a set limit of points you can have, preventing hoarding.
Pathfinder, being the follow-up that rides off the coattails, introduced another optional variant called "Hero Points" in the Advanced Player's Guide. They're effectively identical, though the limit is considerably cut down with certain feats letting you increase that. There are also some spells that involve these Hero Points, but the biggest change here is that they can also be rewarded as an incentive for good roleplay and other OoC good deeds.
4th Edition integrated Action Points into the core rules, but this is in name only. These Action Points are now only provide a free action with certain effects based upon your character's Paragon Path, Epic Destiny, and feats. They're also hard-capped to each player only having one (barring exceptions), recharging only when your character takes an extended rest (making this power effectively daily) or when you hit a milestone by winning enough fights without an extended rest.
5th Edition has Action Points introduced in the Eberron UA, which act similar to how they originally did, but they also have another equivalent called "Inspiration". This is essentially a gift granted for good roleplay and whatnot, and only give a player a chance to roll one attack with advantage (that is, roll it twice and take the better of the two).
In Fantasy Craft and Spycraft, "action dice" can be used for the typical emergency boosts and healing, but they are also required to activate critical hits or turn errors into critical failures, and (in Spycraft) to call in "reserved" equipment when you decide what you need. One set of feats for attack type/skill removes this requirement but takes 3 feats to get to and likely has further requirements. Action Dice are per session (use em or lose em), increase in potency and starting number with level, and can be given by the GM if you do something cool.
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. Every time 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 awarded a hero point. A GM may also give each character a hero point in return for fudging the numbers in the antagonist's favor. 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) without fatigue.
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.
Savage Worlds calls its action points "Bennies". One is given at the start of the session (plus extras for heroes with the Lucky edge and child hindrance) with further awarded for good roleplaying or when the action deck (Savage Worlds uses playing cards for initiative instead of rolling) reveals a Joker for a PC, but are lost at the end of the session. A Benny can be used to reroll, prevent damage, recover, retry imitative, influence the plot or recover power points.
Unusually for an AP system, Savage Worlds gives the Gamemaster Bennies too. The GM starts with a Benny for each player character which can be used freely, plus each enemy wild card (special characters) gets two starting Bennies locked to them and gains one when an antagonist gets dealt a Joker. They can be used the same way to protect antagonists, or