The Fallout series is set in a universe that got stuck in the 1950s for over a century. The transistor was never invented in this universe, so instead of electronics taking off, Fallout is full of techonolgy based on nuclear power, vacuum tubes, and atomic physics. Over the next century and through a course of "Resource Wars" the world is depleted of oil and various countries are dissolved or merged together. This eventually culminates in the Great War, a two-hour exchange of nuclear weapons between the United States, China, and the USSR on 23 October 2077, after which the world was plunged into darkness.
However, not all was lost. The United States, having seen the inevitablity of nuclear war many years before, had set up a number of fallout shelters: deep underground vaults, which would house people until it was safe to come out. These vaults serve as the starting point for many (though not all) of the Fallout games.
Over the years, the Fallout series has changed in genre. The first two games, Fallout and Fallout 2, are top-down, isometric, turn-based role-playing games. Fallout 3, Fallout: New Vegas, and Fallout 4, in sharp contrast, are first/third-person open-world RPGs.
Common to all of the games, however, is the post-apocalyptic wasteland and the enormous freedom they allow to the player. The player character starts out as a blank slate, and you pick out their stats and assign perks depending on what you'd like to specialize in. From there, once you get through the tutorial, you're pretty much free to roam the world and do whatever you like. When talking to other characters, you typically have multiple options to reply, based on your stats, adding to the immersion.
The games have also shifted over time from post-apocalyptic to post-post-apocalyptic. The original Fallout is set in 2161, just shy of 100 years after the Great War, and shows how society has just begun to rebuild. By Fallout: New Vegas in 2281, entire new governments have been established, with all that goes with them.
The first two Fallout games are available only on the PC. Fallout 3 and Fallout: New Vegas are available on the Xbox 360, PS3, and PC. Fallout 4 was released in November 2015 on the Xbox One, PS4, and PC.
There are also several spin-off titles in the series, Fallout Tactics for the PC and Fallout: Brotherhood of Steel for the PS2 and original Xbox. These two games have more or less been declared non-canon; it's complicated. I haven't played either, personally.
More recently, there's Fallout Shelter for iOS and Android, which puts you in the role of Overseer of your very own vault; it's focused around resource management.