The Pureborn, also known as the Enthroned, are descendants of the Qarth's ancient kings and queens, who rule the city of Qarth from the Hall of a Thousand Thrones. The Pureborn also command the city's defenses: the Civic Guard, which includes camelry, and a great military fleet.
There are three principal merchant groups that battle amongst themselves and against the Pureborn for dominance of the city: the Thirteen, the Ancient Guild of Spicers, and the Tourmaline Brotherhood.
To receive an audience with the Pureborn, a petitioner should make a traditional sacrifice in the Temple of Memory, offer a traditional bribe to the Keeper of the Long List, and send a traditional persimmon to the Opener of the Door. If all goes well, the petitioner will be sent a pair of blue slippers, signifying the granting of an audience in the Hall of a Thousand Thrones.
A Clash of Kings
While residing in Xaro Xhoan Daxos’s palace in Qarth Daenerys Targaryen seeks an audience with the Pureborn to plead with them for their fleet of galleys, or part of it and some of their soldiers too. Despite her pleas, the Pureborn refuse her, as Ser Jorah Mormont predicted. Their rejection is adorned with with great courtesy. Mathos says nothing. Wendello praises the way she spoke. The Exquisite refuses her with the rest, but he weeps afterward.
After the destruction of the House of the Undying by Daenerys, the Pureborn's Mathos Mallarawan's wife goes mad and refuses to wear clothes. She feels her silks as insects, crawling on her skin. The woman had once mocked the moth-eaten robe of a warlock. With the various magic practitioners getting their abilities amplified, it was thought that the warlock took his revenge.
A Dance with Dragons
In Meereen Xaro Xhoan Daxos tells Daenerys Targaryen that he went to the Hall of a Thousand Thrones to beg the Pureborn for her life, telling them that she was no more than a child. Egon Emeros replied that Daenerys was a foolish child, mad and heedless and too dangerous to live. 
They listened, but they did not hear, or care … They are Milk Men indeed. They never meant to help me. They came because they were curious. They came because they were bored, and the dragon on my shoulder interested them more than I did.—Daenerys Targaryen, after her visit to the Hall of a Thousand Thrones
The Pureborn see no farther than the walls of Qarth.