In a Quest story, we meet the hero just before he's called on a journey to save the world from oppression, destruction, and all-around evil thingies. Usually, he's sent on the journey after said evil or oppressive force finds its way to his idyllic corner of the world.
Ai is called to his journey in much the same way. The destructive force threatening the planet of Gethen is war. Gethen has never managed a full-on war before, and when we first meet Ai in Karhide, it's clear that such an event is imminent. War can only be prevented if Ai fulfills his quest of securing a treaty between the Ekumen and at least one Gethen country.
In keeping with tradition, he even gets some supernatural cheerleading from Faxe the Weaver. Faxe prophesies that Gethen will join the Ekumen within five years time. When Ai hears this, he's off to complete his mission (although anyone who's ever read a story with a prophecy knows it's never that easy).
Ai's journey really begins when he enters Orgoreyn. A traditional Journey stage consists of battles with monsters and obstacles the hero must overcome. In between these battles, the hero receives help from those supportive of his cause (cough The Lord of the Rings cough).
The Left Hand of Darkness plays with this concept. The obstacle Ai must overcome is the government of Orgoreyn. To do so, he must convince the 33 Commensals to side with him and join the Ekumen. On the other hand, the enemies and allies the hero receives in support of his cause come from the same source, the Orgoreyn government.
Also, in a typical Journey stage, the hero has companions helping him achieve his goal (think the Samwise in, yep, The Lord of the Rings. Or Han Solo in Star Wars, if we want to branch out a little). And, again, The Left Hand of Darkness gives this classic plot a fun twist. Estraven is Ai's companion in this journey. Thing is, Ai doesn't realize the guy is trying to help him. Estraven doesn't travel with Ai like a typical quest companion. Instead, he works behind the scenes to help Ai succeed in his mission.
Finally, the hero is just within reach of his goal, when oops: some new obstacle or opponent comes between him and the prize. Just like in The Lord of the Rings…yeah, you get the idea.
So, Ai thinks the Commensals are ready to side with him. Guess again, Ai old chum. Some of the Commensals have Ai arrested and sent to a forced labor camp. Called the Volunteer Farm, it's the new obstacle standing between Ai and success. There, the guards torture Ai both mentally and physically until his will to live—not to mention his ability to live—is almost broken.
In any Quest story, the hero must undergo a final challenge, the last hard push of the race before the finish line. In The Left Hand of Darkness, the final stretch of the race is quite a long one, 840 miles to be exact.
Estraven arrives at Volunteer Farm and busts Ai out. The two escape into the wilderness. But to get to Karhide, they must hike across 840 miles of the most unforgiving landscape imaginable. It's just like…the planet of Arrakis from Dune only with everything covered in ice and snow instead of sand (thought we were going to say Mordor, didn't you?). The going is tough, but Ai and Estraven managed to survive by growing closer as allies.
The last and greatest ordeal—the final battle, if you will—is the Gorbin Glacier, a.k.a. the Ice. It's the most unforgiving and toughest bit of land on Gethen, and the heroes must cross it to secure their victory.
At the end of any Quest, the hero needs to find his treasure, meet his goal, and/or bring peace. Ai does just that by persuading King Argaven of Karhide to join with the Ekumen. Afterward, peace spreads across the land.
Like for any hero, the journey has changed Ai. When he finally reconnects with his fellow humans, he finds them foreign, alien and unwelcome. So he goes to the only place where he feels like he might fit in: the home of Estraven. There, he tells stories of his adventures to Estraven's father and son.