Symbolism, Imagery, Allegory
Every great science fiction story needs an amazing futuristic gadget. Star Wars has the light saber, Back to the Future 2 has the hoverboard, and Stargate has, um, the Stargate. The Left Hand of Darkness has the ansible, a device allowing the user to instantly communicate across the vast distances of space.
Now, that might not seem more awesome than a ham radio—or a smartphone—but hang on a second. Space is unfathomably huge, meaning any message you send out needs to traverse the vast distances between planets to be received.
If you send a message from Gethen traveling through space at the speed of light (299,792,458 meters per second!), it would still take 17 years to reach the nearest possible destination. Having a device that cuts 17 years down to a couple seconds really manages the lag time between your conversations. (Too bad it'll still take them 17 years to actually deliver your pizza.)
But the ansible is also an important symbol in the novel. As Ai notes:
[The Ekumen] is a form of education; in one aspect it's a sort of very large school—very large indeed. The motives of communication and cooperation are of its essence, and therefore in another aspect it's a league or union of worlds […]. (10.30)
Without communication, the Ekumen can't learn from other cultures and planets or even let them join. And without the ansible, such communication would be impossible. So the ansible basically represents Ai's only means of accomplishing his mission: opening pathways for communication. Neither war nor trade will do, so it's fitting that story's cool sci-fi gadget isn't a light saber or a machine that turns solid rock into chocolate cake or something.
Communication is also important for Ai himself. Without it, he can't complete his mission, as he must learn to open communications, so to speak, between the Gethenians and himself. He uses the ansible twice for this purpose: once with King Argaven and another time with the Orgoreyn Commensals.
Ultimately, each attempt fails as Ai uses the ansible to communicate with them as a member of the Ekumen and not a fellow human being. It is only when the ansible is lost to Ai that he is forced to interact with Estraven as a human and Gethenian, opening the true pathways to communication and completing his mission.
Something to consider: How do the King and Commensals react to the ansible? What does this tell us about their characters and their ways of communicating?