The Sun Throne is a piece of furniture--a chair used as the throne for the monarch of Cairhien. It may also be used as a metaphor for the center of political power in Cairhien.
The chair is located in the Grand Hall of the Sun in the Sun Palace. Some have described it as the soul of the palace. Despite its royal occupation, the chair is a fine example of Cairhienien restraint--though glittering with gold silk and gilt, its plain straight lines and heavy, sturdy arms are what stand out and give the chair its distinctive character. It's only nod to flamboyancy is the large golden Rising Sun affixed high up on the chair's back and made to look as though it is shining down upon the royal occupant's head.
Elayne Trakand, Queen of Andor recently Traveled to Cairhien to lay claim to the Sun Throne. Just prior to seating herself in it, Birgitte, her Warder discovered a small needle, its point coated with a black substance presumed to be poison, set into the chair's cushioning. After its removal, and when questioned whether it was safe for Elayne to sit in it, the irrepressible gaidin responded with, "One way to find out," and to the consternation of both Elayne and the assembled Cairhienien nobility, sat in it, commenting the while, "Not very comfortable."
The metaphor of course refers to Cairhien's chief executive position. Cairhien nobility are adept at playing The Game of Houses and that game in that country centers around the Sun Throne.
Rand al'Thor conquered this city-state and then refused to claim the Sun Throne for himself, instead promising to hold it for Elayne Trakand. Well prior to his having the opportunity to do just that, the White Tower kidnapped him. After regaining his freedom and when he returned to the city, he found that an ambitious noble woman, Colavaere had taken it upon herself to seize the throne and was found seated in it when he entered the Grand Hall. He deposed her, sentenced her to live the rest of her life on a small farm, and returned the fate of the Sun Throne to Elayne.