In Fallout 1 and 2
In the original two Fallouts as well as Tactics and many other spin-off Fallout properties including boardgames, actions were limited by Action Points. Moving, attacking, and opening the inventory all cost action points. The primary influence on the number of action points available to the player was their Agility stat. Perks could also effect action points such as the Bonus Move Perk that gives the player two extra action points exclusively for moving.
One of the most popular new games that incorporated Action Points into their game mechanics was Fallout 3. Since Fallout 1 and 2 both had combat that was entirely turn-based and depended on Action Points, Bethesda didn't want to alienate fans of the originals and found a way to include Action Points and turn-based combat as well as regular FPS style combat. This solution was V.A.T.S.
V.A.T.S is an acronym for Vault-Tec Assisted Targeting System and is part of the Pip-Boy 3000. When in V.A.T.S mode, you can choose what body parts to attack and depending on your weapon, action points will be depleted as you queue up attacks until you can't choose any more. Smaller targets that would usually be more sensitive or cripple the enemy would have a lower percentage of hitting. Some weapons also require more action points than others, for example you can stab someone with the toy knife over ten times but you can probably only fire the Fat Man once before you run out of action points. When you accept your attacks, your character will attack the body parts chosen and if the target dies due to a shot in V.A.T.S, a slow motion camera shot of the target falling will show. In Fallout 3 your character only received 1/10th the damage he would usually take when inside V.A.T.S, making them near invincible when in V.A.T.S. In Fallout: New Vegas Obsidian removed the damage protection from being in V.A.T.S and added special attacks in V.A.T.S that the player can use if they have a high enough skill in that weapon.
Any Fallout game can be played without V.A.T.S at all, but a lot of perks affect V.A.T.S statistics calculation, making your attacks stronger.
There are a lot of different ways to replenish your Action Points, such as Nuka-Cola Quantum or Jet, although overuse of these can cause an addiction. Another perk-based way to replenish your Action Points is through the Grim Reaper's Sprint, which completely fills your Action Points if you kill someone in V.A.T.S. Action Points will also regenerate over time.