If there's one thing you could legitimately complain about in Game of Thrones (apart from all the rapes), it's the bloody names. Jeor this, Wylis that, Aegon, Aerys, Aemon, Malkovich Malkovich Malkovich.
Is it any wonder we can't always keep track of who's who? One name that keeps coming up is Rhaegar Targaryen. He plays a significant part in Game of Thrones, yet we've never so much as seen him. That's because he's dead: Robert Baratheon crushed his chest with a warhammer on the Trident River before taking the throne from the mad king Aerys II long before the events of series one.
But Rhaegar nevertheless played a very important part in the events that followed, and probably will for events yet to come. There are possible SPOILERS below, so now's the time for those who would remain pure to follow a link to something safe.
Who was he?
Crown prince Rhaegar was the elder son of Aerys II, the Mad King, and by all accounts a strong, forceful, charismatic guy. Cersei Lannister was in love with him, but King Aerys felt the Lannisters—despite Tywin being Hand of the King—were beneath Rhaegar, so he was married instead to Elia Martell, the sister of Prince Doran and Sexy Prince Oberyn of Dorne.
(Which explains why the Lannisters killed Elia and the children after the sack of King's Landing, and why Prince Oberyn really hated the Lannisters.)
While Elia was loyal and true—and bore Rhaegar a daughter (Rhaenys) and a son (Aegon)—he was a bit of a turd to her. He fell for Lyanna Stark (Ned's sister) at a jousting tournament and kidnapped her, infuriating her beloved, Robert Baratheon, and precipitating the war of Robert's Rebellion.
The lords who supported King Aerys II in that war did so mostly because even though the king was psychotic, his heir was an altogether better bet.
But Robert beat Rhaegar, so that was that. He had help, of course, from his best bud Ned Stark, and eventually the Lannisters—Tywin turned on his patron by opening the gates of King's Landing to the invading army, and Jaime, of course, delivered the coup de grace by actually killing the mad king, for which he never gets any credit.
What's Lyanna got to do with it?
Rhaegar not only kidnapped Ned's sister, he kept her for months at a place called The Tower of Joy. That was the castle that young Ned and Howland Reed went to in Bran's vision of the past, where they battled Sir Arthur Deyne, the badass knight who fought with two swords simultaneously.
As the fan theory "R + L = J" correctly predicted, Rhaegar got Lyanna pregnant, and Ned found her at the Tower as she was dying in childbirth with baby Jon. He then returned to Winterfell with him, claiming—even to his own wife—that Jon was his own bastard son rather than risk the boy's murder by anti-Targaryen zealots. Such as his buddy Robert, for example, who by then had taken the kingdom from Aerys II.
"So what?", you may be asking. Well, it means Jon Snow has the blood of the dragons, and is therefore potentially a dragon rider like Daenerys. Who, by the way, is his aunt. (Daenerys was Rhaegar's baby sister, born a little after Jon.)
And in all the Targaryen lore, the dragon has three heads. Like Aegon The Conqueror and his two sister-wives with their three dragons, Daenerys has three dragons – but only one rider. So far. Daenerys and Jon would make two, and Tyrion, presumably would be the third, especially if the theory that he's Aerys's bastard son is true.