Dragonstone is the castle that stands upon the eponymous island located in Blackwater Bay. It is the ancestral seat of House Targaryen and the current stronghold of a cadet branch of House Baratheon.
At the time the series begins, it is held for King Robert Baratheon by his brother, Lord Stannis. The Chamber of the Painted Table is inside the castle. The castle was built using advanced Valyrian stonemasonry techniques, long since lost. As a result it has many exotic features unique in all of Westeros, including massive dragon-shaped gargoyles. The entire castle is dominated by dragon motifs and reliefs.
The castle itself is very strong, due to its advanced Valyrian design, which combined with its isolated location make it an ideal redoubt and refuge - first for the Targaryens, and now for its current lord, Stannis Baratheon. Because of these features and its geographic position, it is extremely difficult to breach and can be held by a relatively small garrison against a vastly larger force.
Following the Battle of the Trident and prior to the Sack of King's Landing, Queen Rhaella Targaryen and Prince Viserys Targaryen were sent to Dragonstone for their protection. Queen Rhaella perished during a stormy night giving birth to Daenerys Targaryen.
After King's Landing fell and Robert Baratheon took the Iron Throne, Stannis Baratheon was sent to deal with what remained of the royal family and their retainers. However, Prince Viserys and Princess Daenerys were smuggled across the Narrow Sea to the Free Cities by a loyal knight. Enraged, Robert stripped Stannis of Storm's End and gave it to their younger brother, Renly, and gave Dragonstone to Stannis, to rein in the Targaryen loyalists there. In later years, Robert tried to humor Stannis that he was actually honoring him with lordship of Dragonstone, because under the Targaryens it was the traditional seat of the heir to the throne, but Stannis scoffed at such pleasantries.
Lord Eddard Stark sends a messenger to Dragonstone to personally deliver a letter to Stannis, revealing the secret that Cersei Lannister's children are really bastards born from incest, and thus Stannis is the rightful heir of the dying King Robert.
At the outbreak of the War of the Five Kings, Dragonstone becomes the seat of King Stannis Baratheon. Unfortunately for Stannis, at the outset he is only supported by the lords of the few islands in Blackwater Bay that were already sworn to him. To curry support for his weak position, Stannis converts to the religion of the Lord of Light, influenced by the Red Priestess Melisandre. At a gathering of Stannis and his supporters on the beach, Melisandre burns the wooden statues of The Seven-faced God from Dragonstone's sept, as an offering to the Lord of Light.
A short while afterwards, Ser Davos Seaworth meets with the pirate-lord Salladhor Saan on the coast of Dragonstone, to hire his fleet of thirty ships as sellsails to supplement Stannis' fleet.
Following his defeat at the Battle of the Blackwater, Stannis manages to retreat back to Dragonstone with what little remains of his shattered army.
Stannis and a large portion (possibly even all) his army leave Dragonstone, assisting the Night's Watch in defeating the army of Free Folk at the Battle of Castle Black.
Dragonstone is said to be definitively deserted by the time of Stannis' death.
Whilst discussing their enemies in the Red Keep's new map room, Cersei Lannister ask her brother Jaime where he thinks Daenerys will head first to launch her invasion. Jaime predicts she will land at Dragonstone; it is strategically located with harbors for her ships, Stannis had left it vacant, and it was where Daenerys was born. Jaime's prediction proves correct, as Daenerys returns to Dragonstone around the same time. She spends a few minutes exploring the castle's entryways before reaching the throne room. Declining to seat herself immediately, she proceeds into the Chamber of the Painted Table and declares "Shall we begin?"
Later, Daenerys chairs a meeting with her advisers Tyrion Lannister, Missandei, and Varys during a thunderstorm. She forgives Varys for plotting against her in the past and gets him to reaffirm his allegiance. Daenerys and her entourage are later greeted by Melisandre, who believes that Dany is The Prince That Was Promised. When Melisandre mentions Jon Snow the King in the North, Tyrion convinces Dany that he will make a good ally. Dany accepts on the condition that Jon bend the knee like his Stark forebears. Tyrion sends a missive to Jon Snow.
Dany later chairs another meeting in the Chamber of the Painted Table with Yara Greyjoy, Ellaria Sand, and Lady Olenna Tyrell. Following a heated discussion, Tyrion convinces Daenerys to send Ellaria and the Sand Snakes on Yara's fleet to Dorne to rally the Dornishmen for a siege of King's Landing. Tyrion's plan is to used a combination of Dornish and House Tyrell forces to lay siege to King's Landing to combat the image that Daenerys is a foreign invader. He also successfully proposes sending Grey Worm and the Unsullied to seize the Lannister seat of Casterly Rock. Olenna reluctantly agrees to commit her forces but admits privately that she is motivated by hatred and a desire for vengeance against the Lannisters. Before Grey Worm departs for Casterly Rock, he and Missandei express their love for each other and have sex.
It is possible that after Daenerys and her army march north to fight the army of the dead, the castle is either once again abandoned as it was after Stannis left, or it is manned by a smaller retinue of Stark/Targaryen forces responsible for mining the remaining dragonglass for the war effort against the white walkers.
When the Targaryen family first arrived in Westeros - long before conquering the continent - they made their home at Dragonstone. Built using Valyrian techniques of stonemasonry, since lost to the ages, the castle looks unlike anything else in this part of the world. After Robert took the throne, he bestowed the keep on his brother Stannis.