Rhaegar was the son and heir of King Aerys II, better known as The Mad King, and the older brother of Viserys and Daenerys (Emilia Clarke), who was born after Rhaegar's death. He was popular, intelligent, bookish and a noted musician. Tywin Lannister had planned to marry Cersei to Rhaegar. However, Aerys shot down the idea as being beneath his son. Instead, Rhaegar was married to Elia Martell, Oberyn and Prince Doran's sister. Elia loved her husband and she had two children with Rhaegar, a daughter, Rhaenys, and son, Aegon.
However, as Littlefinger briefly explained to Sansa in Sunday's episode, when Rhaegar won a tourney at the cursed Harrenhal, he passed over his wife Elia and crowned Ned Stark's sister Lyanna as The Queen of Love and Beauty. A year later, Rhaegar allegedly abducted Lyanna, an act which ultimately triggered Robert Baratheon's rebellion. But though the Starks and Baratheons view Rhaegar's actions as a violent kidnapping, according to Targaryen lore, Lyanna loved Rhaegar back and went with him consensually.
As the rebellion against House Targaryen gained steam, Rhaegar and Robert met in combat during the Battle of the Trident and Robert killed the beloved prince. It was the result of this battle that inspired Tywin, who had until then remained neutral, to launch the Sack of King's Landing. During this siege, Jaime Lannister slit Aerys' throat, becoming the Kingslayer, and the Mountain, under Twyin's orders, brutally murdered Rhaenys and Aegon before raping and killing Elia too.