Nearly 300 years after Aegon I's conquest, the reign of his descendant, King Aerys II, spelled the end of Targaryen rule in Westeros. Aerys had hoped to do great things during his reign, but his temper and his descent into madness left the kingdom in disarray — and vulnerable to the machinations of other factions, including the Lannisters and Baratheons.
The final straw came over a romantic rivalry between Robert Baratheon and Aerys's heir, Prince Rhaegar. Rhaegar disappeared with Lyanna Stark, Robert's intended, and both the Baratheons and Starks believed Rhaegar had kidnapped Lyanna. When Rickard Stark, Lyanna's father, and Brandon Stark, her brother, confronted King Aerys about the scandal, he accused them of treason and had the duo executed, igniting a civil war.
Robert killed Rhaegar in battle, and Lyanna's brother, Ned, found her just in time to stay with her as she died. Aerys, now fully the "Mad King," plotted to use wildfire to destroy the entirety of King's Landing — and all of its innocent citizens.
In the end, Jaime Lannister killed King Aerys, and Robert Baratheon took the Iron Throne. Under Robert's rule, any and all Targaryens, whether legitimate or illegitimate heirs, were tracked down and killed. The only ones known to have escaped were Rhaegar's younger siblings, Viserys and Daenerys, who fled to the Free Cities.