Panem is the name of the country where Katniss lives with her family, a country that – after a series of vague disasters – rose from the ashes of North America. Droughts, storms, fires, war: it's all pretty post-apocalyptic. We learn the history of Panem, and the Hunger Games, and the thirteen districts in the first chapter from the mayor. The uprising of the Districts is known as the "Dark Days" (1.74). After the destruction of District 13, the Treaty of Treason was signed to end the conflict, and the Hunger Games were instituted to remind the districts that the uprising must never be repeated.
Suzanne Collins has said that Panem is supposed to be like ancient Rome: "Panem itself comes from the Latin expression 'Panem et Circenses' which translates into 'Bread and Circuses'" (source). This term refers to the techniques used by the Roman Empire to keep the masses happy and docile by keeping their bellies full and their minds entertained (read more here).
District 12 is a very poor coal-mining district located in the region formerly known as Appalachia (3.50). Katniss and her family live in the poor section of District 12, the Seam, where the coal-miners live and work. There is also the Hob, the black market. The district is surrounded by a sometimes-electrified fence from woodlands.
Stephen King calls District 12 "the Chicago Cubs of the postapocalypse world" (source), meaning that they almost never win the Hunger Games. The poverty of District 12 is often contrasted with the wealth of the Capitol.
Speaking of which, the Capitol is a shining city of wealth and grandeur city located where the Rockies used to be (3.50). The people of the Capitol are shallow, speak in a funny, affected accent and value surface appearances, plastic surgery, and entertainment. Being one of the richer districts, they also view the Games as television entertainment. Katniss, of course, despises the place.