The Hunger Games takes place in Panem, which is post-apocalyptic North America. There are several locations in the book and the year is never specified.
The Hunger Games begins in District 12 where Katniss Everdeen lives. District 12's primary export is coal. In the United States, many coal mines are located in the eastern United States, especially Pennsylvania, Kentucky, and West Virginia. It's possible that District 12 is located in one of these areas, though it's never said explicitly.
Katniss and Peeta travel to the Capitol. It's more urban than the districts and people that live there don't work in the same way those in the districts do. It's the center of leisure and the people there dress in bizarre styles. It's the only major city mentioned in the book.
The final setting is the arena for the actual Hunger Games competition. Though it's not possible to know where it's located, the terrain includes hills, forests, and rivers. Katniss says the arenas change every year and mentions that past ones have included deserts and the sea.
Though we don't know what year The Hunger Games takes place, we know that the Hunger Games in the first book are the 74th annual Hunger Games. That means that it's at least seventy-five years after the destruction of North America—and likely much longer, since the system of the Capitol and Districts had to be established and the districts had to rebel before the Hunger Games were ever created.
A fan map made by Entertainment Weekly shows that fans of The Hunger Games believe the Capitol is located in Denver. Check out their placement of the other districts that appear in later books by clicking on the link in References.