Rhaegar Targaryen was the Mad King's son and heir to the throne. Daenerys is Rhaegar's sister.
Lyanna Stark was Robert Baratheon's betrothed and Ned Stark's sister.
Before Robert's Rebellion, the war that dethroned the Mad King and led to Robert becoming king, Rhaegar "kidnapped" Lyanna, or that's the story from the good guys' perspective. At the start of the story we learn that the Targaryens are the bad guys and Robert/Ned are the good guys.
But throughout the series there have been small little hints that the story was wrong. For instance, pretty much every time Rhaegar is mentioned, it's to say what a great guy he was. Basically everything about him was good, even though he was a Targaryen. It always made it seem really weird that he did what he did to Lyanna. He also won a tournament at one point and surprised everyone by (IIRC) giving flowers to Lyanna. Some other hinted things throughout the series (might've even been in book 1) were Lyanna had a big secret that she told Ned. On top of that we hear that Ned is basically this perfect gentleman type, and how extremely odd it was that he cheated on Cat and had a bastard.
So putting all that together the very believable theory came about that Jon was Ned's nephew and that he promised Lyanna he wouldn't tell anyone. And it's a good thing he did, because Robert had every last Targaryen hunted down and murdered. The ones he could find, anyway. In the books, it hasn't been revealed yet. But in the show it was revealed in Season 6 that Jon is Rhaegar and Lyanna's son, through Bran's visions.