This has been tried multiple times, I'm aware, but it's time for try number [reference]42.[/reference]
The OP will focus on the most recent and future titles.

"Fallout is a series of post-apocalyptic role-playing video games created by Interplay Entertainment. Although the series is set during the 22nd and 23rd centuries, its retrofuturistic story and artwork are influenced by the post-war culture of 1950s America, and its combination of hope for the promises of technology and lurking fear of nuclear annihilation."
~Wikipedia
Fallout 3 DLC:
Fallout: New Vegas DLC:
Item Packs:
S.P.E.C.I.A.L.
Strength, Perception, Endurance, Charisma, Intelligence, Luck
The S.P.E.C.I.A.L. system is your character's defining characteristics. It can influence some conversation, effect skills, effect your critical hit chance, and random occurrence. Each primary statistic starts at 5, then you adjust it according to your play style. Each primary statistic cannot go below 1, or above 10, regardless of in game bonuses.
Strength: Modifies hit points, melee damage, and carry weight.
Skills: Melee Weapons
Perception: Modifies minimum distance for enemies to show on the compass.
Skills: Energy Weapons, Lockpick, and Explosives.
Endurance: Modifies hit points, poison and radiation resistance, healing rate, and additional hit points per level.
Skills: Unarmed and Survival
Charisma: Modifies: NPC disposition, and Companion Nerve*
Skills: Barter and Speech/
Intelligence: Modifies # of skill points received per level, dialogue options.
Skills: Medicine, Repair and Science.
Agility: Modifies AP available in V.A.T.S., Weapon holstering speed, and reload speed.
Skills: Guns and Sneak.
Luck: Modifies Critical chance, casino games, random encounters.
Skills: All
Yes, all skills! In New Vegas, Luck effects the skills like this:
Luck of 1 or 2 = +1 all skills
Luck of 3 or 4 = +2 all skills
Luck of 5 or 6 = +3 all skills
Luck of 7 or 8 = +4 all skills
Luck of 9 or 10 = +5 all skills
Combat System
Fallout 1 and 2 uses a turn based system, Fallout 3 and New Vegas use a real-time combat system.
In 3 and New Vegas exists V.A.T.S., or Vault-Tec Assisted Targeting System.
V.A.T.S. is a queuing system. When in combat, hit "V" to activate it. Time slows down dramatically, and allows you to select different limbs to aim for. It shows the chance you have to it each limb.
V.A.T.S. provides a 5% increase in critical hits, but requires Action Points (AP) to use. When in V.A.T.S., you only take 75% of the damage received.
In Fallout 3, V.A.T.S. was significantly better. 15% critical hit, and only 10% of the damage was received. New Vegas nerfed it, however.
Karma System
Karma is the reflection of all actions you've made, both good and bad. The stronger your action is, the more karma points you receive.
Your karma points aren't reflected as numbers. It's just a simple "Good", "Neutral", "Bad" system.
Karma effects what people think of you, and how they respond to you. Some followers require you have a specific Karma level to join you in the Wasteland.
Vaults
Vaults are essentially huge-ass bomb shelters. There were 122 built by Vault-Tec. They can hold 1,000 dwellers in each Vault. The purpose was supposedly to protect citizens, but it was more of a secret Government project, they never actually expected a nuclear war to happen when they built them. Each Vault conducts a different social experiment. You can learn more about each by exploring them in-game. A majority are abandoned, and filled with hostile creatures, crazy dwellers, and hell, even a haunted vault, because you haven't stuff your pants enough yet.
Vaults are protected by giant air-sealed doors. All vaults are maintained and ruled by an Overseer.
PIP-Boy
The PIP-Boy, or Personal Information Processor, is an electronic device developed by RobCo, which goes on your left arm. These where standard issue, and distributed to all Vault-Dwellers. Some received a newer model model than others. They can only be released from the arm by the user, or a skilled technician. PIP-Boys display your health, limb condition, and radiation levels. It's where you equip items, look at the local and global map, and review data received for quests.