|Although this article is based on canonical information, the actual name of this subject is pure conjecture.
The Wight Hunt is undertaken by the King in the North Jon Snow and his allies in order to provide proof to Queens Cersei Lannister and Daenerys Targaryen of the existence of the wights and their masters, the White Walkers, led by the Night King, who had risen again after 8,000 years.
King Jon was accompanied by his principal advisor, Ser Davos Seaworth, wildling chieftain Tormund, Robert Baratheon's bastard son and Davos's friend Gendry, and Ser Jorah Mormont of Daenerys's Queensguard, traveling to Eastwatch-by-the-Sea from Dragonstone. There, they also joined forces with the Brotherhood Without Banners, including Lord Beric Dondarrion, Thoros, and Sandor Clegane, which had marched north from the Riverlands to help play a part in the war against the White Walkers, having been guided by visions. The subsequent fighting at the frozen lake marked the entry of House Targaryen into the Conflict Beyond the Wall and ensured the participation of dragons in the Great War.
At the dawn of the Age of Heroes around 8,000 years ago, a dark and cold winter fell upon the world known as the Long Night, lasting for more than a generation. In that winter, the White Walkers, led by the first White Walker, the Night King, descended upon Westeros for the first time, sweeping through villages and towns and slaughtering all in their path, raising the corpses they made as wights to serve in their army of the dead. Eventually, however, the First Men and the Children of the Forest banded together and defeated the White Walkers in the War for the Dawn. As the centuries passed, both the White Walkers and the Children seemingly disappeared and were assumed by most to have been nothing more than legends and fewer to simply be extinct, with magic gone from the world.
Nonetheless, in the years surrounding Robert's Rebellion and the War of the Five Kings, the White Walkers had risen again with the Night King leading them. They massacred the Free Folk living beyond the Wall, which had been raised by the First Men and guarded by the Night's Watch to prevent the White Walkers from invading southern Westeros again. Though they had been bitter enemies for centuries, the Night's Watch and the Free Folk ultimately banded together to face their common enemy under Lord Commander Jon Snow.
Jon went on to face a mutiny but was resurrected by Melisandre at the urging of Ser Davos Seaworth. With Sansa Stark, Jon defeated House Bolton in the Battle of the Bastards, seeking not only to restore House Stark's control on Winterfell, but also to unify the North to face the resurgent army of the dead when they came. Jon was subsequently declared the King in the North by the lords of the North and the Vale, seceding from the Seven Kingdoms once more and reviving the Kingdom of the North.
However, when Daenerys Targaryen began her invasion of Westeros, King Jon decided to treat with her at Dragonstone, where she controlled the mines of dragonglass, which could be used to kill White Walkers and wights. Daenerys was also the "mother" of three fire breathing dragons that could be used to burn wights in the thousands. After reading a reconnaissance report from Bran Stark, the new Three-Eyed Raven, concerning the movements of the army of the dead, Jon and Tyrion Lannister were ultimately able to convince Daenerys, who had insisted Jon bend the knee to her, to allow the King in the North to travel to Eastwatch-by-the-Sea to capture a wight to use as proof of their return and existence for her and Cersei Lannister, the queen who sat on the Iron Throne and the primary opponent of Daenerys Targaryen in her war.
At Eastwatch, King Jon, who was accompanied by Ser Davos, Gendry, and Ser Jorah Mormont, met with wildling chieftain Tormund Giantsbane to discuss this mission. Tormund brought them down to the ice cells where he held prominent members of the outlaw group the Brotherhood Without Banners prisoner, including the leader Lord Beric Dondarrion, the red priest Thoros of Myr, and Sandor Clegane. Despite their differences and mild mistrust of each other, they all agreed to come together and accompany King Jon and his allies on their mission to capture a wight beyond the Wall.
While travelling beyond the Wall looking for the mountain shaped like an arrowhead as seen in Sandor Clegane's flame-vision, the group interacts with one another. Gendry asks Tormund how he kept himself warm, to which Tormund replies that he did so by walking, fighting, and fucking. Tormund asks King Jon Snow about Daenerys Targaryen, and says that Jon spent too much time with the Free Folk to like kneeling. Though Tormund praises Mance Rayder, their King-Beyond-the-Wall, he wonders how many had to die because he refused to kneel. Thoros asks Gendry if he still hates the Brotherhood Without Banners for selling him to Melisandre, to which he replies that he still does. The Hound tells Gendry that he is whinging, and says that Lord Beric Dondarrion has died six times and never complains of it. Jon offers to return Longclaw to Ser Jorah Mormont, given that the Valyrian steel sword has belonged to House Mormont for centuries, but Ser Jorah tells Jon to keep it and to pass it on to his children. Tormund makes small talk with the Hound and tells him that he hopes to return to Winterfell to find a great beauty, which the Hound deduces to be Brienne of Tarth. Beric expresses his opinion that Jon looks little like his "father," Eddard Stark, who had sent Beric to capture Ser Gregor Clegane at the onset of the War of the Five Kings. Beric recalls hearing that Jon was brought back by Melisandre, and tells Jon he was brought back six times by Thoros. Though Jon does not hold to the Lord of Light, he and Beric both wonder what their purpose is for being brought back. Finally, the Hound sees the arrowhead mountain from his vision, and the group makes their way to it.
Ser Jorah asks Thoros how drunk he was at the Siege of Pyke during the Greyjoy Rebellion, where Thoros had charged through the breach with his flaming sword. Thoros admits that he does not remember. Jorah recalls thinking Thoros was the bravest man he had ever seen, to which Thoros replies that he's "just the drunkest."
Jon Snow and his party are caught in a blizzard. They hear growls in the distance, sending one man to scout the area. After spotting a massive snow bear, he turns and runs back towards the group; as he approaches, Gendry asks if bears have blue eyes. The others see the man running and ready their weapons. Before he can reach them, the snow bear lunges at him from the side, ripping him to shreds. Jon runs after him to try and help, but it's too late. The group forms a circle, weapons at the ready, when they are ambushed by the snow bear, which is in fact a wight snow bear. The monstrous creature mauls and kills one man, swatting Jon aside like a bug when he attacks it. Using their flaming swords, Beric and Thoros light the beast on fire, though it continues to hold out, killing a second man. The Hound, crippled by his fear of the flames, is nearly attacked, but Thoros shoves him aside and is mauled by the bear. Ser Jorah is finally able to stop it when he stabs it with a dragonglass dagger. Beric cauterizes Thoros' open wounds with his flaming sword.
As the group continues, they soon encounter a White Walker leading a small procession of wights. The White Walker stumbles upon burning logs, stopping his group when he is suddenly ambushed. The wights are strong, but when Jon kills the White Walker with Longclaw, all of the wights crumble, save for one which is tied down with a rope with a bag put over its head. Hearing reinforcements coming from the nearby army of the dead, having heard the captive wight's shrieks, Jon sends Gendry back to Eastwatch to send a raven to Dragonstone. Gendry makes it back at nightfall, collapsing at the gate, where he tells Ser Davos Seaworth and several men of the Night's Watch and wildlings to send the raven.
Jon's party runs from the army of the dead; stopping themselves short at the shore of a frozen lake, the group chose to take their chances on the cracking ice rather than certain death at the hands of the oncoming undead. One man was caught by several wights, and the combined weight causes them to crash through the ice into the frigid water beneath, many more wights falling into the disintegrating ice after them. The group manage to reach a small island in the centre of the frozen lake, leaving them temporarily safe from the wights, unable to pursue due to the weak ice. The survivors huddle together for warmth during the night. By morning, however, another of them has passed on: Thoros of Myr has frozen to death in the night. Clegane offers slight comfort by saying, "They say it's one of the better ways to go". At Jon's instruction, Beric burns the priest's body to prevent its return as a wight later.
Ser Jorah stands next to Jon that either the cold will bring them to a slow and freezing death, like what happened with Thoros, or solidifies the lake enough to grant the Army of the Dead passage upon them, but Jon remains adamant in the belief that Dany will come to their rescue and Jorah wonders why the wights fell when the White Walker was destroyed, to which Jon postulates that perhaps that White Walker was the one to turn them; and upon that specific Walker's demise, those resurrected under its power also followed. Jorah suggest that they thin down the hordes of wights by targeting the Walkers, as at least that way, some of the living might survive. But Jon refuses this, reasoning that they came to procure a wight and they did. As for safely getting back with their prize, Jon trusts that by now Daenerys will have, or already has, received Gendry's message and is on her way to provide literal cover fire for the wight hunters' departure; he cites her as their best hope for survival. Beric disagrees and then points his sword to the Night King, flanked by several White Walkers and also mounted on wight horses, who watches over the group patiently. He proposes that they must attack the Night King, that if they manage to slay him, the Great War could end before it even starts, as every one of his created White Walker minions will die with him, as would the wights they've raise from the dead, believing it perhaps to be the reason he and Jon were resurrected. Jon denies this, telling Beric and the party going up head-on against the Night King is out of the question and a suicide mission at best, having done so himself and knows the terrifying encounter all too well.
Out of boredom, the Hound throws a rock across the lake, hitting a skeletal wight and breaking off its decaying, ramshackle jaw. He throws a second stone, which slides across the frozen lake. Realizing that the ice has strengthened, presumably from the presence of the White Walkers, the wights begin to advance on the group, killing at least two men. They are nearly overrun when Daenerys Targaryen suddenly arrives, having departed from Dragonstone, with her three dragons: her mount Drogon, Rhaegal, and Viserion. Multiple strafes of dragonfire not only destroy vast numbers of wights, but also melt huge sections of the lake's frozen surface, helping to limit attacks from the remaining wights.
Jon attempts to hold back the wights as the remaining members of his party climb onto Drogon with Daenerys, the Hound forcing the captured wight onto one of the dragon's horns. With attention focused closer to the group, none appear to notice the Night King aiming a spear at Viserion. With extraordinary strength, he launches the weapon and scores a direct hit, piercing Viserion's throat in a fiery blast. Daenerys watches in horror as the helpless dragon drops like a stone, fire and blood pouring from the fatal wound. He crashes onto the frozen lake, shattering the ice, and sinks below the surface. Drogon and Rhaegal cry in despair for their fallen brother. Daenerys stares on in disbelief as the Night King prepares to throw another spear, this time at Drogon. Jon urges Daenerys to leave just before being tackled by two wights who knock him into the lake. Drogon takes off and is able to narrowly dodge the Night King's second spear, causing Ser Jorah to fall. Fortunately, he is caught by others on the dragon's back and pulled up to safety.
Somehow Jon manages to climb out of the lake. He grabs Longclaw, exhausted and injured, and prepares for a hopeless fight with hordes of wights. After a few moments, though, Benjen Stark arrives, killing several wights and dismounting. Jon recognizes his uncle, who mounts Jon on the horse. Jon urges his uncle Benjen to come with him, but Benjen says there is no time and sends the horse galloping away. Jon looks back as Benjen fights against the wights, eventually being overpowered and killed.
Sandor Clegane brings the captured wight to a nearby skiff at Eastwatch-by-the-Sea. He and Tormund nod out of respect, and Beric Dondarrion tells the Hound that they will meet again, to which the Hound replies that he hopes not. Daenerys Targaryen stands atop the Wall with Jorah Mormont while Drogon and Rhaegal fly around it, crying out for the fallen dragon Viserion. Jorah warns Daenerys that they need to go, but she insists on staying for a bit longer. As she turns around to leave, a horn is sounded for a rider: a nearly unconscious Jon Snow has made it back to Eastwatch-by-the-Sea atop his uncle Benjen Stark's horse. Jon is brought onto a Targaryen ship to rest and recover from hypothermia, exhaustion and minor wounds, passing out on a bed. Daenerys watches on as Davos Seaworth and Gendry strip his frozen clothes for treatment, and sees the wounds that scar Jon's body from his death.
Jon eventually reawakens while the ship sails to King's Landing, finding Daenerys watching over him. He apologizes for going beyond the Wall and for her loss. She explains that her dragons are the only children that she will ever have, and promises to fight the Night King, the White Walkers, and the army of the dead. Jon thanks Dany, which takes her aback, for no one since her brother Viserys has called her that. He then pledges himself to her, addressing her as his Queen, though is unable to formally bend the knee due to the injuries he sustained. She voices concern over what his bannermen will think of this, but Jon says that they will come to see her how he has. Daenerys then leaves to allow Jon to rest, letting go of his hand.
In the far north beyond the Wall, wights utilize massive chains to pull the dead Viserion from the frozen lake. The Night King approaches the dragon, kneels, and lays his hand on it. Viserion's eyes open, revealing the blue glow of all wights and White Walkers.
In the books
In the A Song of Ice and Fire novels by George R.R. Martin, such a mission will not occur. In the Inside the Episode video for "Eastwatch," showrunner D.B. Weiss states that the wight-hunt that Jon is going on is an invention of the TV series, an idea "we came up with" which isn't based on events in a future novel.
For that matter, he says that it's loosely inspired by how Alliser Thorne was sent to King's Landing with the severed hand of one of the two wights that tried to kill Lord Commander Mormont, to present it as proof their claims were real, but it rotted away to nothing by the time he was granted an audience. This did not happen in the TV version - it happened in the books, but was omitted from the TV series (though the way the video is edited, it seems that Weiss was explaining that this drew inspiration from that section of the books, not specifically claiming he thought they did it that way in the TV series as well).