History in the Books
Drogon is first presented to Daenerys alongside his brothers Viserion and Rhaegal as eggs. He later hatches when placed on the funeral pyre of Daenerys' husband, Khal Drogo (Drogon's namesake). When Daenerys tried to immolate herself with Khal Drogo, and sacrificed the witch who had left Khal Drogo in a permanent comatose state, the resulting sacrifice of life for life resulted in Drogon and his brothers hatching, with Daenerys being unharmed by the flames.
As Drogon grew, he was trained by Daenerys, who learned much of dragons by raising him and his brothers. During her time in the city of Qarth, when the Sorcerers known as the Undying steal Rhaegal and Viserion, Drogon goes with Daenerys into the House of the Undying, defending her by burning the pulsating heart that serves as the sorcerers' life source, killing them before she rescues his brothers.
During her bargaining for soldiers in the city of Astapor, Daenerys uses Drogon as payment for 8,000 Unsullied soldiers from the slavers known as the Good Masters. She then uses the Unsullied to kill the Good Masters, after which Drogon returns to her. By this time, he has grown as large as a dog.
During her time in ruling Mereen, Drogon has grown to have a twenty-foot wingspan and roams far and wide, but he has become unruly and dangerous, which has led to him supposedly killing and eating a little girl, with her blackened bones being presented by the child's father as evidence to Daenerys. Drogon disappears when Daenerys tries to chain him up, and only returns when the fighting pits of Mereen are reopened. Drawn by the carnage and the smell of blood, he attacks the pit's fighters and sets much of the area ablaze. A fighter named Herghaz attacks and nearly slays him before Drogon kills him, and Daenerys then, seeing her "child" in danger, subdues him with a whip before mounting and riding him for the first time. Drogon then takes Daenerys away to his lair in the Dothraki sea, where he kills a horse, only for them to encounter Khal Jhaqo, one of Khal Drogo's former Blood-riders. His fate afterwards is unknown.
History in the TV Series
Drogon's history is much the same in the television series as in the books, but he is shown to be far more loyal to Daenerys than in the books, and has grown faster and larger than in the books.
Drogon is again presented as an egg by Khal Drogo as his wedding present to Daenerys alongside Rhaegal and Viserion's eggs. However, Viserys Targaryen, Daenerys' older, mentally unstable brother, tried stealing the eggs, only to be stopped by Ser Jorah Mormont. He hatches in the same way as in the books, placed on the funeral pyre of the dead Khal Drogo, with Daenerys walking into the flames to immolate herself, causing him and his brothers to hatch.
During their time in Qarth, it is Pyat Pree that steals the dragons and places them in the House of the Undying, before luring Daenerys to them to keep his magic strong, as his magic is empowered by the dragons' magic and that they are strongest in her presence. When she is chained up with her dragons, Daenerys unconcernedly whispers "Dracarys", signaling her dragons to breathe fire, and they burn Pyat Pree before incinerating the House of the Undying.
In Astapor, Drogon is traded to the head of the Masters, Kraznys mo Nakloz, for the Unsullied soldiers she needs as an army. When she has their loyalty, Daenerys unleashes the Unsullied on the masters before ordering Drogon to kill Kraznys and then the rest of the Masters.
Drogon and his brothers are also present in Daenerys' tent when a Wise Master from Yunkai arrives to entreat with Daenerys. Drogon, now about the size of a large dog, aggressively barks and snaps at the Master when he gets too close to Daenerys. Later, the Master has two slaves place a chest full of gold near Drogon as an offering to Daenerys. While the two talk, Daenerys throws a piece of meat to Viserion, and Drogon attempts to steal it from his brother. When the negotiation goes sour, Drogon growls at the Slave Master for threatening Daenerys. Scared for his safety, the Master tells his slaves to take back the gold and attempt to leave, Drogon stops them in their tracks, hissing and growling, forcing them to leave empty-handed.
By the time of Season 4, Drogon is the size of a small horse, and during a confrontation between him and his brothers, Drogon snaps at Daenerys when she tries to settle the dispute, causing her alarm that she is losing her influence on them as dragons never truly can be tamed.
Following her conquest of the city of Mereen, Drogon, now the size of a small ship, roams the countryside and kills a herd of goats. He later kills a young girl, leaving her father with the child's blackened bones. This forces her to chain up Rhaegal and Viserion. When he reappears to Daenerys, he is now the size of an orca, and she attempts to touch him, only for the dragon to fly away at the last moment.
As he flies west, Drogon is seen amidst the ruins of the Valyrian Peninsula by Sea Jorah Mormont and Tyrion Lannister, causing Tyrion to become temporarily speechless.
While Daenerys attends the grand re-opening of the fighting pits at the Great Pit of Daznak, a large group of Sons of the Harpy, rebels who seek to return Mereen to the tradition of slavery, appear and attack her council and the spectators. While trying to escape, Daenerys and her small group become surrounded by the Sons with no chance of escape. As she is about to accept her imminent death, Drogon suddenly appears, flying out of a flame burst and begins to attack, burning the Sons of the Harpy with his fiery breath and tearing them to pieces with his powerful jaws. Temporarily retreating in terror, the Sons of the Harpy regroup and retaliate by throwing spears at Drogon, wounding him, but when Daenerys commands him to fly away, he does so with her on his back.
Later, Drogon lands atop an impressive hill in a sea of green grass, which serves as his lair in the Dothraki Sea. Though Dany tries to convince him to take her back to Meereen, Drogon, still recovering from his wounds, is uninterested in flying back. Daenerys wanders away to find something to eat, but she is quickly surrounded by a Dothraki khalasar.
After Daenerys takes over command of the Dothraki, she senses Drogon following her. She slips away and rides him around the khalasar before landing and giving a rousing speech to her khalasar mounted on his back. He is significantly larger than he was during the battle in Meereen, being at least 90-100 feet long with a similar wingspan. Eventually, Drogon carries Daenerys to the Great Pyramid of Meereen, which is under siege by the Masters.
Daenerys discusses terms of surrender of the masters in Meereen, but corrects them by saying the terms are for their surrender, not hers. The masters foolishly mention slaughtering the dragons during the dictation of terms, which angers their mother. Drogon answers Daenerys's call and lands next to her so that she can mount him. As Drogon takes flight, Viserion and Rhaegal break out of the catacombs in which they were chained up to join their brother in the sky at last. At Dany's command of "dracarys", Drogon leads his brothers in immolating the slaver fleet with dragon fire, decisively ending the Masters' siege of Mereen, allowing her to take the enemy fleet for their own. Drogon then, alongside Rhaegal and Viserion, fly with Daenerys, her army and fleet of ships to Westeros at last.
By Season 7, Drogon is roughly the size of a 747 Jumbo Jet.