A few definitions for the theory:
- The ‘world’ is Westeros
- The ‘darkness’ — or the Long Night — is the Targaryen reign
- ‘Azor Ahai‘ is history’s song of Ice and Fire, personified at the end as Jon Snow
The first attempt: Tempered in Water
“It was a time when darkness lay heavy on the world … And so for thirty days and thirty nights Azor Ahai labored sleepless in the temple, forging a blade in the sacred fires. Heat and hammer and fold, heat and hammer and fold, oh, yes, until the sword was done. Yet when he plunged it into waterto temper the steel it burst asunder.”- A Clash of Kings, Davos I
The Targaryens first brought the darkness by invading the riverlands (e.g. water). They landed at Blackwater Rush and were subsequently challenged by several riverlands houses.
Even when certain early-stage battles were not in the riverlands (such as the battle at Gulltown), the battles took place in water.
The first attempt - ‘tempered in water’ - reached its conclusion at Harrenhal in the riverlands, near the God’s Eye - perhaps the eye of R’hollor, the Lord of Light, the one who sent Azor Ahai.
Harrenhal was a turning point in Aegon’s Conquest. Other major battles were still to ensue, but Harrenhal was now infamous.
It is at Harrenhall that 'Azor Ahai' now lays in waiting. R'hollor choses its hero and names the hero the Song of Ice and Fire.
Waking Azor Ahai: The Tourney at Harrenhal
For hundreds of years, the Targaryens brought darkness to world, while the chosen hero - the song of Ice and Fire - lay dormant at Harrenhal.
Then the Tourney at Harrenhal happened.
Here’s a succinct excerpt from A Wiki of Ice and Fire (emphasis mine):
Lord Walter Whent hosted a great tourney at Harrenhal in 281 AC. The famous tournament resulted in the crowning of Prince Rhaegar Targaryen as its champion, and is also noted for the participation of the mysterious Knight of the Laughing Tree. Ser Jaime Lannister was named to the Kingsguard of Aerys II Targaryen during the tourney. Prior to the outbreak of Robert’s Rebellion, Rhaegar abducted Lyanna Stark less than ten leagues from Harrenhal.
The famous tourney at Harrenhal brought together the needed notes of Ice and Fire to awaken Azor Ahai and end the Long Night.
- Tournament takes place at Harrenhal, where Azor Ahai - the song of Ice and Fire - was first defeat, and left in waiting
- Rhaegar wins the tournament — Fire
- Knight of the Laughing Tree (popularly theorized to be Lyanna Stark) — Ice
- Rhaegar crowns Lyanna the queen of love and beauty — Ice and Fire
- Jaime Lannister is named to the Kingsguard — A Lion is born
The tourney at Harrenhal effectively sung the first verse of A Song of Ice and Fire, awakening the prophesied hero on the very ground in which he was last defeated.
The tourney also set into motion all future events related to the prophecy and leads to the end of the Long Night.
The second attempt: A Lion’s heart
“The second time … Azor Ahai captured a lion, to temper the blade by plunging it through the beast’s red heart, but once more the steel shattered and split.”- A Clash of Kings, Davos I
Azor Ahai, or the Song of Ice and Fire, had chosen Jaime Lannister - and, by extension, the Lannister family - as the second attempt to defeat the darkness and end the Long Night.
Jaime Lannister was chosen at the tourney at Harrenhal. The ‘Kingslayer’ killed the Mad King, Aerys II Targaryen, before the Aerys could burn King’s Landing to the ground.
Other lions (Tywin Lannister) marched on King’s Landing and attempted to kill the entire Targaryen lineage and end the darkness. However, some Targaryens escaped.
The Targaryen family plotted for years from beyond the Narrow Sea. And so, even with no Targaryen on the Iron Throne, the darkness continued its reign.
The third attempt: Azor Ahai, Nissa Nissa, and the Dawn
A hundred days and a hundred nights he labored on the third blade … he summoned his wife. ‘Nissa Nissa … bare your breast, and know that I love you best of all that is in this world.’ She did this thing, why I cannot say, and Azor Ahaithrust the smoking sword through her living heart … Such is the tale of the forging of Lightbringer, the Red Sword of Heroes.- A Clash of Kings, Davos I
By the third attempt to bring the Dawn, Azor Ahai - the Song of Ice and Fire - was personified as Jon Snow, the child of the Ice (Lyanna Stark) and Fire (Rhaegar Targaryen) that had been awoken during the tourney at Harrenhal.
Even though Jon is actually a Targaryen by name, he is the perfect balance of Ice and Fire and (in the show) has rejected his Targaryen lineage. Therefore, he is not considered part of the 'darkness' that the Targaryen reign represents.
Jon Snow (Azor Ahai) and Daenerys Targaryen (the last of the darkness) fall in love, and Jon is forced to kill her to save the world from the darkness, fulfilling the prophecy and bringing the dawn.