As a lot of people know by now, Daenerys Targaryen is almost certainly the aunt of Jon Snow. But if you're still a little confused by the whole thing, this post is for you.
Let me take you back to the time Aerys II Targaryen, aka the Mad King, was on the Iron Throne.
Aerys II was married to his sister, Rhaella*, with whom he had three kids: Rhaegar, Viserys, and Daenerys.
*Yes, the Targaryens are huge fans of incest, believing it important to keep the bloodline "pure". It's why they look so ~unique~ and some of them have a tendency towards madness. They also practiced polygamy – Aegon the Conqueror, Daenerys's ancestor who invaded and united the Seven Kingdoms, married both his sisters. In the books, Daenerys grew up believing she'd marry Viserys.
Rhaegar was a lot older than his siblings (he actually died before Daenerys was born). He was married to Elia Martell, sister of Oberyn. Rhaegar and Elia had two kids: Rhaenys and Aegon.
Elia nearly died having Aegon, and subsequently could no longer have kids. Elia apparently loved Rhaegar, and in the books he's described as being "fond" of her.
Meanwhile, Lyanna Stark, sister of Ned, was betrothed to Robert Baratheon.
In the books, Lyanna doesn't seem particularly happy about the engagement, which was arranged between her father and Robert. She was aware that Robert had already fathered a bastard, and so she didn't believe he'd be faithful to her.
These characters collided at the Tourney of Harrenhal. This has been mentioned only briefly in the show, but here's what we know from the books: At Harrenhal, Howland Reed (father of Jojen and Meera) was bullied by some squires, and Lyanna Stark stood up for him, tended to his wounds, and took him to a feast.
At the feast, Rhaegar – who was a talented musician – played a beautiful, melancholy song that made Lyanna cry.
During the tournament, a mysterious knight, known as the Knight of the Laughing Tree, challenged and defeated the three knights whose squires had bullied Howland. Most people believe this knight was Lyanna in disguise.
Rhaegar ultimately won the tournament, and crowned Lyanna the "queen of love and beauty" instead of his wife, Elia. We saw Littlefinger talk about this incident on the show in Season 5.
The following year, Rhaegar and Lyanna ran off together. There are a few options for how this went down.
Brandon Stark (Lyanna and Ned's older brother) rode to King's Landing and demanded Rhaegar's death. Aerys II accused Brandon of treason and arrested him. He then summoned Rickard Stark (Brandon, Ned, and Lyanna's father), who was arrested when he arrived in the capital.
Aerys II then demanded the heads of Robert and Ned from Jon Arryn, the Lord of the Vale who the pair had lived with for some time (it was his death that Ned was investigating back in Season 1). Jon Arryn, Ned, and Robert all called their banners, and Robert's Rebellion began.
In order to forge alliances, Ned married Catelyn Tully (who had previously been betrothed to his brother Brandon), and Jon Arryn married Catelyn's sister, Lysa.
Ned and Catelyn were together only briefly after their marriage before he went off to fight in the war. But they conceived Robb in that short period of time.
One of the major battles during Robert's Rebellion was the Battle of the Trident, where Robert and Rhaegar met in one-on-one combat.
Rhaegar injured Robert, but Robert dealt Rhaegar a fatal blow with his war hammer. It's said that Rhaegar died with Lyanna's name on his lips.
Meanwhile, Aerys II sent his wife, Rhaella (who was pregnant with Daenerys), and Viserys to Dragonstone for their own safety, but he refused to let Rhaegar's wife Elia or her children leave King's Landing.
Until this point, the Lannisters had stayed out of the fight. When Tywin appeared at the gates of King's Landing with his army, declaring his loyalty to Aerys II, the Mad King let him in. The Lannister army proceeded to sack the city.
Aerys II ordered Jaime Lannister, who was part of his kingsguard, to bring him Tywin's head. He also ordered the wildfire he'd had stored around the city to be set alight. To save the lives of thousands – and his own father – Jaime killed the Mad King.
Meanwhile, the Mountain*, fighting for the Lannisters, killed Rhaegar and Elia's children, and raped and killed Elia.
*In the books, Amory Lorch kills one of the children, but that hasn't been mentioned in the show.
Ned arrived in the city to find it sacked by the Lannisters, and Jaime sitting on the Iron Throne, having killed his king. When Robert arrived later, he and Ned had a huge argument over the deaths of Elia and the Targaryen children. Robert was pleased about it, but Ned was disgusted, and left the capital.
Ned ended up in Dorne with some companions (amongst them, Howland Reed), where he came across some knights of the Targaryen kingsguard, including Ser Arthur Dayne, at what Rhaegar had dubbed the Tower of Joy.
We saw this in one of Bran's visions on Season 6 of the show. It's significant that at this point, Aerys II and Rhaegar were dead, yet the Targaryen kingsguard were still guarding something. That something being Rhaegar's last remaining child – our boy Jon Snow.
While many believe Jon is still a bastard, because Rhaegar was already married before he kidnapped/ran away with Lyanna, there are two popular theories that suggest ways Jon could be legitimate. The first is that Rhaegar had his marriage to Elia annulled and then married Lyanna (EDIT: THIS ACTUALLY HAPPENED – see update below), meaning Jon would be the next heir to the Targaryen throne. The other theory is that Rhaegar remained married to Elia but also married Lyanna, continuing his ancestors' practice of polygamy. Once again, this would still give Jon the strongest claim to the throne.
As we know, Ned and his companions fought the kingsguard, and killed them. Howland was injured, but survived. Ned entered the tower to find Lyanna – who had been missing the whole Rebellion – dying after having given birth to baby Jon.
We don't know Jon's real name yet, but there's a good chance it's very Targaryen (Ned named him Jon after Jon Arryn). Before her death, Lyanna asked Ned to protect her baby, because Robert would want him dead – something that would have stood out all the more in Ned's mind after having witnessed Robert's pleased reaction to the deaths of Rhaegar's other children.
Ned returned home to Winterfell and presented Jon as his bastard son to protect him. Catelyn, who had given birth to Robb in the meantime, was not impressed (to put it mildly).
Robert was crowned king in King's Landing, while at Dragonstone, Rhaella had already declared the young Viserys king, since all the other Targaryens were dead (to her knowledge). Later, during a great storm that destroyed the Targaryen fleet, Rhaella gave birth to Daenerys before dying herself.
So there you have it! Jon is the son of Lyanna Stark and Rhaegar Targaryen (NOT Robert Baratheon, as some people think – seriously, the timeline doesn't add up). Which makes Daenerys his aunt, even though she's a little younger than him.
What does that mean for their relationship?
We will have to wait and see how it all plays out, but one thing is for certain...
UPDATE
Season 7, Episode 5 confirmed that Rhaegar had his marriage to Elia annulled, and secretly married Lyanna. Meaning JON IS LEGITIMAGE. OMG!
Read more about it here.