- "Robert's Rebellion was built on a lie. Rhaegar didn't kidnap my aunt or rape her. He loved her, and she loved him."
- ―Bran Stark
Rhaegar Targaryen and Lyanna Stark began their relationship as secret lovers, yet eventually married in a private ceremony. Most in Westeros believed that Rhaegar had kidnapped Lyanna, who was betrothed to Robert Baratheon, and raped her, but they had in fact loved each other. They ran away to Dorne and lived in the Tower of Joy. They are the true parents of Jon Snow (born Aegon Targaryen), who is the true heir to the Iron Throne.
Before Robert's Rebellion
To cement the relationship between the Iron Throne and Dorne, Rhaegar was wed to Elia Martell, sister of Doran and Oberyn Martell. With Elia, Rhaegar had two childern, a daughter, Rhaenys and a son, Aegon. However, Rhaegar never found true happiness with Elia.
Rhaegar's and Lyanna's relationship began at the Tourney of Harrenhall, about one year before Robert's Rebellion. Rhaegar was one of the combatants in the tournament, and eventually he made it to the final round, where he faced off against his friend and personal bodyguard, Ser Barristan Selmy of the Kingsguard. Rhaegar won the round and was awarded with the crown of Winter roses. He was then supposed to give it to the most beautiful maiden in the crowd. What Rhaegar did with it shocked everyone. He silenced the entire audience by giving the crown to Lyanna Stark, rather than his own wife Elia. And the fact that Lyanna was engaged to Robert Baratheon, the act was most disrespectful to him. It's not fully known whether Rhaegar and Lyanna had been planning to elope together before or after the tournament.
About a year later, Lyanna ran off with Rhaegar to Dorne. Before this, Rhaegar had secretly annulled his marriage Elia, without her knowing. After the annulment, Rhaegar and Lyanna wed in secret in a Faith of the Seven ceremony before a Heart Tree, in the presence of the Seven and the Old Gods (their respective religions). After the wedding they settled in the Tower of Joy, where they lived happily. At this point both Rhaegar and Lyanna had abandoned their arranged marriages and defied everyone who wanted to use their names to secure their own houses.
However, Rhaegar's and Lyanna's actions had very severe consequences to their respective houses, as well as the entire realm. Lyanna's eldest brother Brandon Stark, accused Rhaegar of kidnapping her and he traveled to King's Landing to demand Rhaegar's father, Aerys, to surrender Rhaegar for judgement and return Lyanna. Instead of handing over his son, Aerys had Brandon imprisoned for treason. Alerted by this, Lyanna's father Rickard Stark went to King's Landing himself to demand his son's release, as well as Lyanna's. This time, however, Aerys burned Rickard with wildfire and executed Brandon as well. Aerys then demanded that Lyanna's second brother Eddard and Robert to be handed to him as well. This act crossed the line and Jon Arryn raised the banners against Aerys, with Eddard and Robert joining him, beginning Robert's Rebellion.
At the beginning of Robert's Rebellion, Rhaegar and Lyanna first remained at the Tower of Joy. But as the tide was turning against the Targaryens, Rhaegar eventually marched to war with his loyal followers, leaving Lyanna to the Tower under the guard of Gerold Hightower and Ser Arthur Dayne.
Rhaegar fought in the Battle of the Trident where he was killed personally by Robert Baratheon, as the latter wanted revenge for Rhaegar for kidnapping and raping Lyanna, which never really happened. With the loss of Rhaegar, the end result was nearly certain. Afterwards, House Lannister proceeded to sack King's Landing, resulting in the deaths of Rhaegar's first wife Elia and their two children, Rhaella and Aegon as well as Aerys, who was stabbed in the back by Jaime Lannister. After the war was over, Eddard Stark continued his searching for Lyanna, eventually finding her in the Tower of Joy. He killed Rhaegar's remaining loyalists and rushed to the tower, where he found Lyanna dying of childbirth, having just given birth to Rhaegar's son. In her last moments, Lyanna whispered the name of her newborn son to Eddard: Aegon Targaryen. She also wanted Eddard to promise her that he will protect him, as Robert would surely kill him, if it came to be known that he is Rhaegar's son. Eddard promised to protect him and passed of the baby as his own bastard son, Jon Snow, keeping his true identity secret from everyone, even Jon himself.
Oberyn Martell, the brother of Rhaegar's first wife Elia, comes to King's Landing for the first time since Rhaegar and Elia's wedding many years before, seeking vengeance for his sister's murder. During his visit, he talks about the marriage of Elia and Rhaegar, and states that Rhaegar "left her for another woman", implying a belief that Rhaegar and Lyanna's relationship was consensual and not a case of Rhaegar taking Lyanna away against her will.
When Sansa Stark is returned to Winterfell by Petyr Baelish, she goes to the WInterfell crypts. Petyr finds her next to the statue of Lyanna, and she laments how her father Ned never talked about her. Petyr then reveals how he saw Lyanna once at the Tourney of Harrenhall and witnessed Rhaegar giving the crown of Winter Roses to Lyanna instead of Elia. Sansa then says what she believes is the truth about Rhaegar kidnapping Lyanna and raping her. Finally, Petyr cryptically remarks how thousands had to die, because Rhaegar chose Sansa's aunt.
In a vision, Bran Stark sees his father arrive at the Tower of Joy, to find Lyanna having just given birth to Jon Snow, confirming that she is his real mother, and that his real father must not be Ned but someone else instead.
During his training in the Citadel of Oldtown, Samwell Tarly was transcribing old writings. Gilly was reading one such document, which said that "Prince Ragger" had his marriage annulled, so he could marry another. However, Samwell was too frustrated to be interested. However, after coming to Winterfell, he met with Bran, who had discovered the truth about Jon's origins through his power of Greensight. Bran claimed that Jon is the son of Rhaegar Targaryen and Lyanna Stark, and that his last name is in fact Sand, not Snow. Samwell then realizes the importance of Gilly's discovery: Jon isn't a bastard at all, since Rhaegar and Lyanna had wed before he was born. Bran uses his powers and finds Rhaegar's and Lyanna's wedding and concludes that Robert's Rebellion was built on a lie, and that Jon (whose real name is Aegon Targaryen) is the true heir to the Iron Throne.