Well obviously fans know that Jon Snow knows nothing. But does Jon Snow know he's related to Daenerys on Game of Thrones? Um, probably not. Jon has been dealing with the mystery of his birth for his entire life and as of right now, it looks like Bran is the only person in Westeros who knows who Jon's real parents were. As a bastard, Jon was never technically entitled to any of the same rights and privileges as his siblings, and had tried several times to ask Ned Stark about his mother. Basically, being raised a bastard is a large part of Jon's character.
Through Bran's visions, Game of Thrones viewers know that the popular fan theory that Jon's parents were Rheagar Targaryen and Lyanna Stark is true. Because Rheagar was Daenerys' brother, she is actually Jon's aunt. In the vision, Bran watched Howland Reed and Ned Stark fight Rheagar's Kingsguard at the Tower of Joy, after Rheagar was killed by Robert Baratheon. The soldiers were there to guard Lyanna Stark, believed to have been kidnapped and raped by Rheagar. There, Lyanna gave birth to Jon and died in the process, but not before making Ned swear a promise to her.
She made Ned promise to take care of Jon and to keep his true parentage a secret for his own protection. After the rebellion, all Targaryens were in danger, and only Dany and her brother managed to survive. This is why Ned never told Jon who his mother was. Tragically, before going south in the first season, Ned told Jon that the next time he saw him he'd tell him about his mother. Of course, they never did see each other again, so Jon never learned the truth. Neither, it should be noted, did Daenerys.
Of course, now that things are speeding up and Jon and Dany are already working out some trade deals, I have a feeling that the truth is going to come out at some point this season. The secret of Jon's parents died with Ned until Bran saw it in his vision, which means that Bran is the only person who knows the truth. Now that he's at Winterfell, Bran may finally get the opportunity to tell Jon. If I were Bran, I'd wait to deliver this news in person, as it's not the sort of thing someone generally wants to hear in a letter. However, it doesn't look like Bran is picking up on social cues these days, so who knows.