Since the very first season of Game of Thrones, we've known Jon Snow is a bastard. It's pretty consistent fuel for ridicule and embarrassment in the earlier episodes, but it never stops him from rising to greatness. He may be a bastard, but he's Ned Stark's bastard, and that commands respect. For all intents and purposes, Jon Snow is part of House Stark, and he has acted as such. Then, in season six, everything changes. Through one of the many flashbacks of Bran Stark, the new Three-Eyed Raven, we learn that Jon Snow isn't Ned Stark's bastard at all. At the same time, Game of Thrones confirms a long-held theory: that Jon Snow's real parents are Rhaegar Targaryen and Lyanna Stark. It may seem inconsequential at first glance. After all, Lyanna Stark is Ned's sister! Jon's still half a Stark like always! In truth, though, the revelation changes everything.
At this point, it's crucial to consider how the monarchy system works in Westeros. It's a lot like the medieval ages: it's a patriarchy, with an emphasis on men and male heirs. This means, since Jon Snow's father is a Targaryen, he's technically more Targaryen than Stark. That also means he's entitled to the Iron Throne over Daenerys Targaryen, who thinks she's the only member of her house still alive. But she's wrong. Jon Snow is a Targaryen.
The interesting thing to note is that no one knows Jon Snow is a Targaryen, not even Jon himself. That's why Jon's the King of the North, and that's why everyone sees him as a Stark. Even Bran, who learns part of the truth in the flashback, only knows that Ned is not Jon's father, and Lyanna is his mother. Of course, with the truth floating around somewhere in Westeros, it's only a matter of time before someone discovers that Rhaegar is his father. And we have a few ideas for how it could play out. Once that secret comes to light, everything else will change.