Although nothing's been said yet in the show, it's fairly clear that everything takes place in an alternate version of Earth.
So here's a look at their world.
This is the map shown in the opening credits. They say that the world is divided into four nations; the Water Tribe, Earth Kingdom, Fire Nation, and the Air Nomads. With this map, we can get a feel for where everything takes place and even keep track of where Aang is going.
We can see that the Fire Nation is the western red continent; the most northern and southern blue continents are the Water Tribe's home; the large Europe/Asia-looking continents is the Earth Kingdom; and the white islands in the northwest, south, and east are Air Nomad territory.
Individual Places and Landmarks
Listed by location -- North to South
