Who was the first avatar in the Avatar (the last airbender) universe? How did they achieve their powers? And why is there only one?
This past weekend had a Korra 2-part episode titled Beginnings, which answers this question. It followed the life of Wan, the first Avatar. 10,000 years prior to Korra's time, Wan lives in a city on the back of a lion-turtle. The lion-turtle grants citizens of the city temporary firebending power when they leave the city to hunt. Wan joins a hunting party and gets the firebending power. He then sneaks back into the city, bypassing the required relinquishing of the firebending power. When he's caught by the city's authorities, he's banished, but the lion-turtle allows him to keep the firebending power.
While banished to the wilderness, he befriends the spirits that live there, and hones his firebending power from the crude summoning of bursts of fire to something that resembles modern firebending. In so doing, he becomes more skilled at firebending than the hunters, and he uses this to defend the spirits from the hunters' attacks. He eventually decides to leave the area and find other lion-turtle cities.
During his journey, he comes across two powerful spirits battling. He uses his firebending to break up the battle between the spirits - Raava and Vaatu - and in so doing Vaatu flees. Raava explains to Wan that interrupting the battle was a terrible mistake. Raava is the force of light and peace. Vaatu is the force of darkness and chaos. She had restrained Vaatu for the past 10,000 years, containing evil and chaos.
Every 10,000 years, Raava and Vaatu have a battle to determine who will control the fate of the world. Vaatu starts increasing his power by corrupting spirits, turning them evil. This has the dual purpose of weakening Raava as well, by diminishing the power of good. Doing so then will ensure that Vaatu will win the battle, and spread evil and darkness across the world. The 10,000 year cycle is due to end in one year.
Wan attempts to gather more bending powers so he can battle Vaatu to atone for unleashing evil onto the world. He comes across a lion-turtle who grants airbending, and requests the power of airbending. The lion-turtle explains that no human can hold two element bending powers at the same time, he gives the power to Raava, who will hold it until Wan can master it. Raava and Wan train, and eventually Wan gains airbending. They continue traveling, and gain water and earth bending. As they do, Wan's skill and power continues to increase.
Eventually the year is up and so Wan, Raava, and Vaatu do battle. Raava is incredibly weak at this point, and permanently merges with Wan, further increasing his power. Wan bends the four elements and uses them to seal Vaatu away in a tree, thus winning the battle. He then becomes the first Avatar, and swears to bring peace to the world while bridging the human and spirit worlds. The story jumps forward to show an elderly Wan on a battlefield near death. He rues his failure in bringing peace to the world, and then dies, thus starting the Avatar cycle.
Note - Many details of the episode are omitted or streamlined here for the sake of brevity.
At this point in time, we have not been told in any canonical resource what the name of the first Avatar was, or where he got his powers, although we do know that he was from one of the Waterbending Tribes. What we do know of the beginnings of the Avatar universe as we know it gives very little indication as to what actually caused the Avatar cycle to begin in the first place.
We do know from the Lion Turtle in the episode Sozin's Comet, Pt. 2: The Old Masters, that in the era before the Avatar, the only bending art was Energybending. From the episode The Firebending Masters, we learn that the original Airbenders were Energybenders who watched the Sky Bison bend the air to fly, and that the original Firebenders learned how to focus their energy into fire by watching the Dragons. Likewise, in the episode The Siege of the North, Pt. 1, we learn that Waterbenders learned by watching the Moon exercise control over the tides. The original Earthbenders, Oma and Shu, learned their art by watching the Badgermoles shift the earth away to dig a tunnel through the mountains, as told in the episode The Cave of Two Lovers.
From what we know from canon, we can hazard a fairly solid guess at several things. Likely, the familial tribes of Energybenders spread out from a central origin point. Since they all had the latent ability to bend the elements, they were able to learn from the native animals of the region they spread to, or in the case of the Water Tribes, from the moon itself. Living on the Poles would contribute to that, since they are said by Sokka to have a full season of night. We know from the Lion Turtle that once the Avatar Cycle started (the Era after the Avatar) Energybending was a lost art. From that syntax, we can conclude that the Avatar did not exist before the Four Bending Arts had diverged. The Avatar Cycle begins with Water, so we know that the first Avatar was a member of one of the Water Tribes. Likely, as one of the Avatar's roles is the Bridge Between the Worlds, the first Avatar discovered his powers by being shown by the Spirits, and being told of his role as the Balance between the Four Nations. Presumably while journeying to learn the other three elements, the first Avatar would share his story with the people he met. The proof is in the pudding, or so they say, and most people would be convinced by someone wielding more than one element simultaneously. The rest is history.
The Avatar Cycle is never explicitly explained on the show, therefore, there is no easy citation. The origin of the Avatar Cycle lies in the origins of the traditional Chinese calendar, which has been in use for thousands of years. The Chinese year starts on the second new moon after the Winter Solstice. Traditionally, the Chinese seasons each correspond to one of the four basic elements. Winter represents Water, spring represents Earth, summer represents Fire, and Fall represents Air. The Avatar Cycle has been seen in the show's succession of Avatars to mimic this order, as well (Yangchen to Kuruk, Kuruk to Kyoshi, Kyoshi to Roku, Roku to Aang, Aang to Korra). Since the Avatar Cycle mimics the traditional Chinese seasonal calendar, it can be inferred that it also begins with Water.