A decade or so after the fall of the dynasty, Viserys Targaryen attempted to build support to retake the Iron Throne. He secured a key alliance by marrying his teenage sister, Daenerys, to Khal Drogo, a Dothraki warlord.
Daenerys's early days of marriage were traumatic and frightening, but in time, she came into her own and developed a real love relationship with Khal Drogo. When Viserys threatened a pregnant Daenerys, Drogo had him killed with molten gold.
In quick succession, Daenerys lost both her unborn child and her husband. She placed her dragon eggs on Drogo's funeral pyre and walked into it herself, only to emerge unharmed with three newly hatched dragons.
Over the years, Daenerys took up the quest for the Iron Throne herself. She freed thousands of Unsullied slaves, took over Meereen, and built up an impressive army. It's at this point in the story that the books and TV series diverge. "House of the Dragon" takes place in the TV universe, so we'll go from there.