He seemed to appear to Daenerys at some points before his birth in dreams, the first time prior to her wedding to Drogo and shortly after her brother Viserys had beaten her, towering over Dany before she awoke from fright. When she was having difficuly adapting to her role of Khaleesi, even going so far as to contemplate suicide, he again appeared, towering over her again and covered symbolically in Targaryen blood. He stared Danerys down before dousing her in fire, "cleansing" her of her old life.
When he was born, many of Daenerys' followers believed him to be the second coming of Balerion the Dread, the largest and most powerful of the Targaryen dragons at the time of the Conquest. Daenerys believed it would be suiting that she give him a new name to both honor the dead and give him a new chance in life.
Covered in black scales, Drogon is the largest and fiercest living dragon in all the world, larger than his brothers Viserion and Rhaegal. Half matured, Drogon killed and ate a child near Meereen. When Daenerys locked her dragons away to keep her people safe, Drogon flew away to the Dothraki Sea.
When the fighting pits of Meereen were re-opened, Drogon appeared in Daznak's Pit, drawn by commotion and the smell of blood and death. The animal handler Harghaz attacked him with a spear, but was killed by Drogon, who then flew away with Daenerys on his back.