The Chamber of the Painted Table is the main hall in the castle of Dragonstone. It is named after the table that dominates the room, carved in the shape of Westeros and engraved with its major cities and landmarks. The chamber is accessed through a passageway located just behind the right hand of Dragonstone's throne.
The chamber is where Stannis Baratheon holds meetings with his court, including Maester Cressen and the Red Priestess Melisandre. After receiving Eddard Stark's message concerning the true heritage of King Robert's children, Stannis dictates a letter to his scribe in the Chamber of the Painted Table to be sent to lords throughout the land revealing this information and therefore establishing his claim to the Iron Throne as the rightful heir. Maester Cressen tries to poison Melisandre in the chamber to remove her influence on Stannis, only to die from the poison himself while Melisandre is unharmed.
Melisandre seduces Stannis on the table, which leads to the birth of the Shadow she uses to assassinate Stannis' brother and rival Renly Baratheon. After his defeat at the Battle of the Blackwater, Stannis returns to Dragonstone and confronts Melisandre in the Chamber of the Painted Table about the validity of her predictions. He begins to strangle her in fury but relents when she reminds him of the spell they used to kill Renly. He experiences remorse of murdering his brother. Melisandre warns him that he will commit worse betrayals before their long war is over but insists that he must fight on. She shows him a vision in the flames that awes him and restores his faith in her.
After Daenerys Targaryen's fleet arrives from Meereen, her first act is to seize Dragonstone, as both a symbolic gesture (the location of her birth and the ancestral home of House Targaryen) and a strategic one (deep ports to moor her ships and proximity to King's Landing). After exploring the castle, Daenerys and her Hand, Tyrion Lannister, arrive in the Chamber to prepare for the invasion of the rest of Westeros, as Aegon the Conqueror once had. The following day, Daenerys grills Varys on his true loyalties, comparing his desire for a Targaryen restoration to that of Viserys, in which she also states that Varys only supported her when it suited him, and followed Viserys until his death. She also questions why he betrayed her father for Robert, to which he explains that he would have been executed if he had not done so and that he obeyed Robert's strength in contrast to Aerys' cruelty. Varys then makes it clear that he is truly a representative of the common people. Daenerys then requests that Varys make a promise to advise her when she goes wrong, rather than betray her, to which he concedes.
Later, Tyrion plans the conquest of Westeros. There, Daenerys later stops the arguing between Ellaria Sand and Tyrion over the assassination of Myrcella Baratheon, to which Daenerys replies Ellaria must respect her Hand. She also agrees with Tyrion over Yara's idea to attack King's Landing immediately, and that the Unsullied should attack Casterly Rock while the Westerosi armies lay siege to the capital. When everyone has left the chamber, she requests an audience alone with Olenna Tyrell. Daenerys then tells her that she knows Olenna is on her side due to their mutual hatred for Cersei, rather than a love for Daenerys herself. In response, Olenna encourages Dany to be a dragon, rather than a "sheep", like the other high lords and ladies.
Later, at the Chamber of the Painted Table, Jon tells Daenerys about the news of his half-brother Bran and half-sister Arya Stark's return to Winterfell. He warns Daenerys' about Bran's vision of the Army of the Dead marching towards Eastwatch-by-the-Sea. Tyrion is present and proposes bring a Wight south in order to prove that the Army of the Dead and the White Walkers are real. Varys opines that it is suicide trying to appeal to Queen Cersei Lannister but Tyrion argues that he can persuade his brother Jaime.
Davos also thinks such a mission is risky even for a smuggler like him. Jorah volunteers to go north to help capture a Wight while Jon volunteers to lead such the expedition. Apparently on the verge of tears about the idea of Jon leaving, Daenerys responds that she did not give Jon permission to leave but Jon reminds her that he is the King in the North. He tells her she has the power of life and death over him but that he trusted her even though she was a stranger. He pleads with her to return the favor by trusting him.
In the books
In the A Song of Ice and Fire novels, the Chamber of the Painted Table is located at the top of the central keep of Dragonstone, called the Stone Drum. It has four tall windows facing each point of the compass. Aegon the Conqueror planned his invasion of the Seven Kingdoms from the chamber. The Painted Table itself is more than fifty feet long: roughly twenty-five feet wide at its widest point and four feet at its thinnest. At the precise location of Dragonstone is a raised seat that allows the occupant to view the entire map. The series omits this raised seat, with Stannis sitting at the head of the table near Dorne and the Reach (the Wall can be glimpsed at the opposite end of the table from where Stannis sits). It's location has also been changed, as it now seems to be located further down in the keep and has only a single window taking up an entire wall, carved directly into the cliff face.