Several generations after the civil war, Aerys II Targaryen (later known as "The Mad King") sits on the Iron Throne with his eldest son, Rhaegar, as his presumed heir. While he was once a generous and benevolent king, he soon begins to unravel mentally due to several factors: his incestuous lineage (he's the son of brother and sister, Jaehaerys and Shaera Targaryen, who themselves come from a long, incest-filled line), the incessant loss of family (he witnessed the deaths of three sons), and being held prisoner for six months during the Defiance of Duskendale uprising. All the while, his paranoia grew, and he descended into uncontrollable madness. He was ultimately killed by a member of his own Kingsguard, Ser Jaime Lannister, during Robert's Rebellion, making him the 17th and final Targaryen to rule on the Iron Throne.
The Mad King appears briefly in season six, episode six, of "Game of Thrones" in a blink-and-you'll-miss-it flashback, portrayed by David Rintoul (pictured above).
