The Dragon is the title for the champion of the Light against the Dark One, and is always the reincarnation of a single particular soul. He is reborn in the Age of Legends and again in the Third Age (and equivalents along the Wheel) to combat the Shadow. It is unknown if he is born again in a similar repeating pattern in other Ages. The Dragon of the Age of Legends was Lews Therin Telamon. The Dragon of the Third Age is Rand al'Thor. The Prophecies of the Dragon as described in the Karaethon Cycle give vague impressions on the events that will follow after his coming.
What Elan Morin Tedronai says in the Prologue of TEOTW indicates that Lews Therin Telamon only earned the name Dragon during the War of Power, so as Elan Morin Tedronai was named Betrayer of Hope. Lews Therin Telamon was never mentioned to be named Dragon Reborn.
In addition, the word dragon is also used to describe a kind of red/gold serpentine reptile, which is often displayed on the banners and other identifying trappings of the human Dragon. It is not known if this creature ever existed in reality, or if it simply a legendary beast that the Dragon is named after.
Rand al'Thor, the reincarnation of Lews Therin as the Dragon Reborn, is fated to both destroy and save the world. The Pattern requires that he be a male channeler and must be at Tarmon Gai'don or the forces of the Light will not stand a chance; even his presence does not assure victory, however, only the possibility of one. Prophecy says that he would break all bonds and bring suffering to the earth. It is commonly expected that in doing so he will also bring a new Breaking of the World. Due to Lews Therin Telamon's infamy, for better or for worse, the tale of the Dragon Reborn is known well even in remote villages, though often distorted beyond recognition. There are also villages or lands where even talking about the Prophecies is forbidden or avoided. E.g. Emond's Field. That's why Rand, Mat, Perrin, Egwene and Nynaeve know nothing about the Karaethon Cycle and cannot put pieces together, cannot recognise events happening around them until they learn about the Prophecies. 
Every social group appears to have their own name for the Dragon in the 3rd Age. For example, the wolves know him as Shadowkiller, the Sea Folk know him as the Coramoor, and the Aiel call him Car'a'carn, the chief-of-chiefs He Who Comes With The Dawn. Seanchan people know him as Dragon Reborn, similar to Westland people, although they have different prophecies about him.
Men who have declared themselves to be the Dragon Reborn but do not fulfill any prophecy are said to be false Dragons. This is often done for fame or glory, but in the end all have fallen from power. The first of the false Dragons to become of notice was Raolin Darksbane in 335 AB, who managed to assault the White Tower before he could be stopped.
False Dragons were rare until Gorin Rogad declared himself in 995 NE. Then three years later, in 998 NE, Logain Ablar rose to power. Three more rose in 998 NE, though of these three only Logain Ablar and Mazrim Taim were capable of channeling.
Ishamael has accused the White Tower of controlling most, if not all of the false Dragons. In 999 NE, Siuan Sanche, former Amyrlin Seat of the Tower accused the Red Ajah of doing the same thing with the help of the then-gentled Logain's testimony. While it is believed to be merely a ploy to rouse support against the Amyrlin Elaida's rule, it has been hinted multiple times that there may be some truth to it.
It is written that the pattern creates false dragons, decreasing the time interval in between them as the time gets closer and closer to the real dragon being born as can be seen between the large time gaps between Raolin and Yurian and then suddenly the pattern spits out Logain, Taim and the other nameless false Dragon before Rand declares himself. It is also written that when the Dragon Reborn is declared, the pattern has no need for false dragons which is why when Rand finally accepted himself as the Dragon Reborn at Falme, all of the other false dragons fell to some phenomenon at the time.
The true Dragon of the Third Age was revealed to be Rand al'Thor in 998 NE. Since then, many of the prophecies have been fulfilled. Two former false Dragons have acknowledged him as the true Dragon, and Tarmon Gai'don appears to be fast approaching.
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