At the end of Game of Thrones Season 6, the show finally answered the question that book readers have been harboring since oh, say around 1997 or so: Who is Jon Snow's mother? Though they didn't actually show him, they also answered the other very important question: Who is Jon Snow's father? Jon Snow is a Targaryen, born of Rhaegar Targaryen and Lyanna Stark. But where does he fit on the family tree? With the two of them obviously attracted to each other, the new question is: How exactly are Daenerys and Jon Snow related?
I've seen people refer to Dany and Jon as brother and sister, cousins, etc. This is not that surprising. The show has drilled into us that Targaryens married brother to sister, and that incest is really not a huge deal in that family. It's natural for fans who aren't paying that close attention to who the Mad King is, versus who Rhaegar is, to assume that Jon and Dany will end up turning out to be long lost siblings, or that their similarity in age means they're first cousins.
But that's not how it works. Let's break down the Targaryen Family Tree a bit and explain.
Jaehaerys II Targaryen
Jaehaerys was the second son of Aegon V "Egg" Targaryen. (Yes of the Dunk and Egg series, and brother to the late Maester Aemon at the Wall.) He did not originally assume he would become king. A sickly boy, he only lived to age 37. He ruled as the 16th Targaryen King, from 259 - 262 AC.
Aegon was against the practice of incest and originally engaged Jaehaerys to a Tully. But he was in love with his sister, Princess Shaera. Following in the footsteps of his older brother Duncan the Small, who broke his own engagement to a Baratheon to marry a commoner, Jenny of Oldstones, the two siblings eloped. Duncan forfeited the crown, making Jaehaerys the heir, and Aegon had to break another engagement. (This strained the Targaryen relationship with the Baratheons and Tullys very badly, long before Robert's Rebellion.)
Aerys II "The Mad King" Targaryen
After Jaehaerys became king, his son Aerys wanted to marry for love outside the family. But his father forced him to marry his sister Princess Rhaella. Jaehaerys had a prophecy told to him, via a witch brought to him by Jenny of Oldstones, that his son and daughter's line would yield "The Prince that was Promised." Aerys never loved Rhaella, but he worked tirelessly to have as many children as possible, three of which survived: Rhaegar, Viserys, and Daenerys.
Aerys did not start out mad, and when he took the throne in 262 AC, there was hope his would be a time of peace. We know how that went. One of the factors blamed for his mental collapse was the miscarriages, stillborns and early deaths of the children that Rhaella had over her life, which is why there are decades between her three that lived to adulthood.
Prince Rhaegar Targaryen
By the time Rhaegar was born, the ability of trueborn Targaryens to give birth to healthy children had decreased substantially. Rhaella kept miscarrying, so he had no sisters, and there were no available Targaryen women of age to betroth him to. There was a mighty game played by many, attempting to have their daughters engaged to him. (Including Tywin Lannister trying to get Cersei as Rhaegar's wife.) But in the end, he was married off to Dornish princess Elia Martell.
They had two children, Rhaenys and Aegon, both of which nearly killed her. She was told after that not to have any more children. Rhaegar believed it was fine, as he was convinced Aegon was "The Prince that was Promised."
Then, in the year 280 AC, during the Year of the False Spring, there was a great tourney at Harrenhal, which some believe Rhaegar had worked to make happen so he could meet with the other lords about removing his now very unstable father from power.
Instead, Rhaegar met a very nice girl from a very nice Northern house who was very unhappily engaged to Robert Baratheon. He lost his head so badly, he named her the Queen of Love and Beauty after winning the whole damn thing, even though his wife was sitting right there.
As we know now, Lyanna Stark was not kidnapped, but went willingly with Rhaegar from Winterfell after that. She got pregnant and was kept in the Tower of Joy for safety reasons as the war turned against the Targaryens.
Meanwhile, at Dragonstone, Rhaegar's mother Rhaella was also pregnant with yet another child of Aerys. She and Lyanna Stark would give birth within days of each other, both dying in child birth as their side of the battle was lost.
Dany Is Jon's Aunt
So there you have it: despite being the same age, Dany is actually Jon Snow's aunt.
As for Jon's real name, many believe it will turn out to one of two things. Either it will be Jaehaerys, for the last sane king of the Targaryen line, or Aegon, because Rhaegar decided that would be the name of "The Prince that was Promised." (This theory surmises in his wild love for Lyanna, he transferred the certainty of it being the Aegon he had with Elia to the son he was having with her, and named two of his sons the same thing.)
Also of note: Jon is still most likely a bastard, unless papers are found that prove otherwise. (Remember, Elia was killed by the Mountain. Rhaegar could have hurried off and remarried with Lyanna before his son was born.) The Tower of Joy is technically Dornish territory, so even though the Targaryens are from the Crownlands, his last name won't be Waters. Bastard names are by location born, so that would make him Jaehaerys Sand or Aegon Sand.
At least, until Daenerys decided her future husband should be made legit so she can marry him. She has the power to do that, you know.