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.

Earth Kingdom Base
This base is located in the western Earth Kingdom.

Built among the mountains on the western coast of the Earth Kingdom, this base is one of the first stops for returning soldiers from the war. It has an infirmary, housing, pens/barns for the ostriches, and open spaces for training. There also seems to be chambers underneath the base. The outer circular wall protects the area, while the thin walls branching out give easy access to outside areas. In the center of the base is a six story pagoda, and the fifth floor is the office of General Fong. He is a powerful Earthbender who rules the base, and wants to use Aang's avatar state as the ultimate weapon against the Fire Nation in order to end the war.

Points of interest: Well, it's a nice base, and its wide areas are good for training either physical or earthbending.

Village Resort
This village is located along the coast somewhere in the northwestern Earth Kingdom.

Gorgeous in the spring, this village is surrounded by cherry blossom trees and natural beauty. A lot of the main buildings and spa areas were built on top of the river, and condo-like houses line river's banks. This area is also close to the ocean, and even has a dock. Uncle Iroh and Prince Zuko stayed here for a short while after drifting on the sea for three weeks. Considering Iroh and Zuko had very little besides the clothes on their backs, it's possible they got to stay here for free.

Points of interest: Such a perfect location to relax and recharge. The village is a dream land with beautiful surroundings, friendly people, massages and free robes.

Northern Water Tribe
The Northern Water Tribe is located on the northern continent.

Hidden within the North Pole, the Northern Water Tribe is an incredible contrast to the decimated Southern Water Tribe. Thanks to the treacherous landscape and strength of the tribe's many warriors & waterbenders, the Northern Water Tribe has been able to survive the war so far. The inner city is built from thick walls of ice and the intricate water streets are very similar to the Italian city of Venice. There seems to be plenty of food, good economy, and the culture is very intact. Some of their traditions include that women aren't allowed to learn waterbending, and they probably aren't allowed to fight in the war either. However, women in the tribe who can waterbend are sent to Yugota to learn how to use their gift for healing. Another tradition is the engagement necklace - instead of a ring, a man will carve something into a blue pendant that will be worn by his betrothed. There are also arranged marriages.

Points of interest: The city is gorgeous, and the culture is... interesting. There's plenty to see and learn about, and there's also adorable creatures, like the koala-otter. ^-^

Northern Air Temple
The Northern Air Temple is in the Earth Kingdom mountains.

Built on a mountain high above the clouds, the Northern Air Temple is practically in a world of its own. Long ago it was the home to airbending monks, as well as the championship for Air Bison Poling. But time and war caused the Air Nomad's disappearance, though new people - refugees from a flood, eventually found it many decades later. Building upon what they found, an inventor "upgraded" the temple with pipelines pumping hot air and made many inventions, including hang-gliders inspired by the temple paintings. But these upgrades destroyed the historical remains of the Air Nomad society. Not to mention the utter desecration of building Fire Nation weapons inside the temple's sanctuary room... Horrible.

Points of interest: Inside the temple are remains of walls covered with paintings and mosaics of the Air Nomad's history, as well as statues of previous monks and airbenders. The outside of the temple is lined with Lionhead Bison stone statues. There's also hang-gliding for the adventurous, cuddly hermit crabs, and if you're not careful, large explosions. With the high concentration of natural gas inside the mountain, one must be careful where they bring a fire.

Abandoned Earth Kingdom City
In the northern Earth Kingdom is an old city and a herbal institute.

What once appeared to be a thriving city built into a mountain, is now crumbled from abandonment. A large stone carving of an anteater or something accompanies the road that enters the city. Inside some of the buildings you can find tapestries and pottery, scrolls, and even a crown among other things.

On a nearby mountaintop rests a herbal institute, one of elegant architecture and past wealth. For 40 years a herbalist made a living here, and watched the other residents slowly leave. Now only she and her cat (Miyuki) remain, tending to wounded Earth Kingdom soldiers when they come by.

Points of interest: Everything! Abandoned civilizations are so much fun to explore. ^_^ The herbalist's greenhouse is very nice too, and the valley swamp is good for frozen frogs.

The Great Divide
This canyon is the largest in the world.

In the northeastern region of the Earth Kingdom lies the Great Divide -- the largest canyon in the world. "Experts" say that the canyon was probably formed by earth spirits who were angry at the local farmers for not offering a proper sacrifice. .... There is no shrubbery or water inside the canyon, and bringing food is dangerous because it attracts Canyon Crawlers - giant ant-like creatures that could -and would- easily swallow you whole.

Apparently, it is necessary to cross this canyon in order to reach the capital city of Ba Sing Se, and takes about a day or two to cross on foot. Many of the paths leading into and out of canyon are sheer cliffs or have giant gaping holes preventing further travel. Luckily, the Great Divide Tour Guide is an Earthbender, so he can easily create stone paths and bridges long enough for those with good intentions who need to cross.

Points of interest: ...Well, it is the largest canyon in the world.

Fire Temple / Crescent Moon Isle
This crescent moon island may be located in the central ocean.

Isolated within the Fire Nation ocean, this crescent moon shaped island was the home to the Fire Temple for Avatar Roku. Although the history of the temple is relatively unknown at this point, Avatar Roku use to live there, so the temple was at least 100-200 years old. However, recently when Aang visited the temple, the events that followed that day resulted in the temple being destroyed. And as irony would have it, it was Avatar Roku himself who destroyed the temple with the aid of the island's active volcano. With the temple sages "escorted" by Zhao to the mainland, the island is thought to be deserted.

Points of interest: Uh... none! With Avatar Roku's temple destroyed, the only thing that remains is the volcano.

City of Omashu
The City of Omashu is located in the southern Earth Kingdom.

Located... somewhere in the Earth Kingdom continent hidden amid a mountain range is the City of Omashu. Built atop a large platform in the middle of a canyon, the only way to enter the city is by means of flying or the singular stone trail leading up to it. Omashu is a beautiful and very large city filled with civilians, Earthbending guards, a mail delivery system, and more. A wise (but slightly crazy) Earthbending king named Bumi leads them. Because the city is so large, they've developed an intricate system of delivering supplies throughout the city by way of chutes and slides with the aid of earthbending. It's quite the spectacle.

Points of interest: The city's architecture, sliding down the delivery chutes in a cargo box, the restaurant, elevated areas where you can see the amazing view, the King's palace... there's a lot to choose from.

Kyoshi Island
Kyoshi Island is located somewhere in the southern ocean.

Somewhere in the southern islands lies Kyoshi Island, named after a female Avatar that once lived there centuries ago. Although the current residents don't appear to have bending skills, the female warriors are quite skilled and their outfits and weapons mimic that of the ancient Avatar. Kyoshi Island had stayed hidden from the war before Aang showed up, and has a decent fishing market. Unagi and Elephant Koi reside near its bay, and the population is fairly large for a hidden village.

Points of Interest: The tall statue of Avatar Kyoshi near the entrance to the village, Elephant Koi surfing in the bay, and the Kyoshi warrior training dojo.

Southern Air Temple

The Southern Air Temple is located in the Potola Mountain Range down south.

Located deep within the Potola Mountain Range north of the Southern Water Tribe continent is the Air Nomads' Southern Air Temple, accessible only by means of flying. What once use to be a beautiful thriving home to flying bison, winged-lemurs, and lots of airbending monks is now a desolate shell of what it once was. This is also where Aang lived and trained before the iceberg incident. It also was home to Monk Gyatso, the greatest airbender and Aang's mentor.

Points of interest: The Air Temple Sanctuary, where the statues of previous Avatars are kept; and the airball field. It's a sport involving a hollow ball, airbending, and not a fear of heights. There'll be a video game on it one day I'm sure.

South Pole

The South Pole is located on the most southern continent.

The southern most continent is home to many wintry animals and what remains of the Southern Water Tribe. The Fire Nation in their quest for world domination destroyed what use to be a magnificent city of ice. Since then, the tribe has lived in igloos and tents hidden in the snow in attempt to stay under the Fire Nation's radar. This is where Katara & Sokka lived before they left to help Aang.

Points of interest: Penguin Sledding. And... the abandoned Fire Navy ship left over from the first wave of attack.