The reign of Viserys I was fraught with questions about who would succeed him, which "House of the Dragon" depicts. Viserys's first marriage only left one surviving heir, his daughter Rhaenyra, who he declared his intended successor. His remarriage after his first wife's death, however, produced three sons and a daughter. Viserys's insistence on keeping Rhaenyra as his heir led to the formation of rival factions at court.
House Targaryen, as "Game of Thrones" pointed out, had a practice of intermarriage within their own family line. When Rhaenyra was widowed, she married her uncle, Prince Daemon, whose own wife had died, and the couple had three children.
Upon the death of Viserys I, his widow Alicent and the commander of the Kingsguard, Ser Criston Cole, chose to defy his stated wishes for the succession, and crowned Viserys and Alicent's eldest son as King Aegon II. Rhaenyra was not even aware of her father's death until later.
Rhaenyra refused to give up the throne and declared war on her half-brother Aegon II. The resulting civil war became known as the "Dance of the Dragons" and permanently weakened House Targaryen. Nearly all of the dragons in existence were killed during the brutal civil war, as well as many key members of House Targaryen. With so many heirs now dead, Aegon II was forced to betroth his surviving daughter, Jaehaera, to Rhaenyra and Daemon's son, Aegon the Younger, who eventually ascended the throne as Aegon III.