The year is 2277. 200 years ago China and America's war over fuel went nuclear, resulting in the end of the civilized world. Before the bombs landed many people made it into giant underground vaults scattered throughout the United States where they, and their descendants, lived out their lives safe from the Atomic Bomb's initial and after-effects. Some people managed to survive on the surface, others left the vault sooner rather than later, and eventually human re-emerged onto the earth and tried to survive in the apocalyptic wasteland left behind by atomic warfare.
There's a few coalitions and armies left behind, the Enclave and the Brotherhood of Steel for one, but for the most part everything is very local. Some local governments have managed to emerge, especially via the use of a G.E.C.K. (device for re-establishing civilization). The west is especially civilized with the New California Republic and Vault City. Fallout 3 however, is set in the east around Washington D.C. where there's only a few towns that have a local government which doesn't stretch beyond the tiny borders of their dilapidated settlements.
In Fallout 3 you play a character who emerges from a vault 200 years after the apocalypse, who gets involved with a project to better the "capital wasteland."
EDIT: Other thing worth noting, Fallout 3 is an alternate timeline, not our own. The exact point of departure remains relatively unknown, but in Fallout 3 the United States culture never surpassed beyond the Red Scare mentality. Paranoia about communism, 1950s style ads, and war-time propaganda were essentially the norm in the Fallout-Universe United States before the bombing. They also had a more "family values" oriented culture, it seems.
Science is also different on a fundamental level from our laws of physics; they developed cars with mini-nuclear reactors, plasma/laser weapons like in the old sci-fi movies, mini-nuke launchers that creature mushroom clouds the size of a tree etc. This manifests itself by mutated creatures of a wide variety in the aftermath of the bombs too. From gigantic ants to mutant people who look a lot like Frankenstein, and zombie-like things known as ghouls; the post-apocalyptic world is full of freaks.