Water
Earth
Fire
Air
The four elements are very important to the Avatar world, they are the staple that keeps everything together. Each element corresponds to their respective nation: Water Tribe, Earth Kingdom, Fire Nation, Air Nomads. I'll speculate about the nations later, but for now I'll stick to the facts.
Manipulation of the elements in the Avatar world is called bending, so Waterbending, Firebending, etc.
Water is the element of change, the people of the Water Tribe are adaptive and very strong in their community. There are many different styles of Waterbending, Southern style, Northern style, even swamp style. In the opening credits, the Chinese character meaning water is paired with the character for goodness or virtue. These characters certainly speak for the element of water, which brings life and healing. The strongest of Waterbenders can change water between the three states (Liquid, solid, and gas.) According to Princess Yue of the Nothern Water Tribe, the first Waterbender was the moon. The ancient people saw how it pushed and pulled the water and learned how to do it themselves. Water and Fire are similar in the aspect that the origins of both were a solar object, more on that later.
Earth is the element of substance, the people of the Earth Kingdom are strong and brave. In the opening credits, the Chinese character for earth is paired with the character for strength or durability. Toph says that Earth is a stubborn element, it takes a strong person to be able to control it. The origins of Earthbending are much like Airbending, they both came from observing animals. In the episode "Cave of Two Lovers" it is revealed that the first Earthbenders Oma and Shu learned from Badgermoles who made their homes in the mountains. Toph later explains that Badgermoles are blind, but they learned to see with the earth.
Fire is the element of power, the people of the Fire Nation are strong willed and able to complete anything they put their mind to. Normal fire burns red, but extremely talented benders such as Azula can bend blue fire. Fire and Air are the only two elements that can be created from nothing. Both Earth and Waterbending require a source to bend. In the opening credits, the Chinese character for fire is paired with the character meaning intensity. Fire is a strong element controlled by emotion. Like Water, which can be manipulated to be water, ice, or vapor, Fire can change its form to become Lightning. Again, it takes a very powerful bender like Azula to change fire into this state. The original Firebenders learned to Firebend by observing the ancient dragons. Like Waterbenders who draw their power from the moon, Firebenders draw their power from the sun. The original Firebenders were the Sun Warriors, implying that they learned to Firebend not just from the dragons, but also from the sun.
Air is the element of freedom, the people of Air Nomads detached themselves from daily living to reach enlightenment, and they apparently had great senses of humor! In the opening credits, the Chinese character for air is paired with the character for peace and harmony. Air is a very important element in the show, as Aang, the main character, is the last Airbender alive, hence the shows title. The people of the Air Nomads are Monks, who have reached enlightenment and inner peace by detaching themselves from the world and worldly concerns. Like Fire, Air can be created from nothing. The original Airbenders learned to bend from the now extinct Sky Bison, or Flying Bison, but that's not the point. A central theme of Avatar is the war between the Fire Nation and the rest of the world. Because the Air Nomads had no formal military, they were easly wiped out by the Fire Nation during the first attack by Fire Lord Sozin.
That's Avatar 101. My next post will be about the characters. I'll do that when it's not so late. It's currently 12:10, and I can literally sit for hours ranting about the characters, which I know I'll do at this hour.
There were several quotes from the show in this post, if you can pin two of the non-obvious ones, I'll give you a shout out in another post.