This post is from Reddit, simple background on Rhaegar and the popular theory that he's Jon Snow's father. The post reaches a bit with prophecies at the end, but the background is explained nice and simple which I thought some people here might like.
If you are only a fan of the show, Rhaegar's identity and actions are murky at best, so for those wanting a little background, I threw this together real quick. If you have read the books, this is all old news to you. This post contains heavy book spoilers regarding other characters on the show, not just Rhaegar.
Rhaegar was born during the Tragedy at Sumerhall. Sumerhall was exactly what it sounds like, a summer vacation spot for the Targaryen royal family. Now the details are sketchy, but the rumor is the Targs were trying to hatch dragon eggs through some sort of magical ritual that involved, you guessed it, a lot of fire. Whatever they were doing it went horribly wrong, and the fires burned down most of Summerhall, and a large portion of the Targaryen family with it. It was during this event that Rhaegar was born.
At a young age, Rhaegar was very bookish, without the least bit of martial inclination, leading many to joke his mother ate books when he was in the womb. One day, after reading an unknown tome, Rhaegar (R from here on out) marched up to the master at arms in the Red Keep and said something to the effect of "It seems I need to be a warrior. Train me." From that point on, R was one of the most respected warriors in all the in the Seven Kingdoms, rarely losing a tournament, and being held in awe by such notable BAMF's as Barristan Selmy and Jaime Lannister.
Now R never lost the melancholy that his birth left on him, and would often return to the ruins of Sumerhall alone. He constantly kept a very unique silver harp with him, and was known to play so well as to leave the ladies in tears. Basically, anything R did, he was one of the best in the world at.
R was the first born son of the Mad King Aerys Targaryen, the man would later murder Ned Stark's father and older brother Brandon. After Aerys refused to marry R to Cersei Lannister, he was instead wed to Elia of Dorne (the younger sister of Prince Oberyn, the Red Viper.) Elia gave him two children, who were later murdered by the Mountain in the sack of King's Landing. "You raped her! You Murdered her!" That was Rhaegar's wife, Elia and their children.
Along with Rhaegar, the Mad King also had two more children, Danerys and Viserys. But Rhaegar was the first born son, so any succession would go through him and his children. This is very important if you believe the theory behind Jon's true heritage, as it would mean his claim would come before Dany's, but we'll get back to that in a second.
Ok now that family trees are out of the way, after R and Elia had been married for a bit, there was a great tournament at Harrenal, the one mentioned by Littlefinger in the episode with Sansa in the crypts of Winterfell. So anyways, after winning a tournament, which R did, typically the victor would crown a woman as "The Quenn of Love and Beauty." This was a ceremonial title of course, and in this case R crowned Lyanna Stark, as opposed to his wife Elia. So this was were most people point to when the trouble began.
We don't ever get told exactly why R did this, but it's assumed he was privy to some type of prophecy that foretold of a dragon with three heads that would save the world, presumably from the Others. R already had a son, Aegon (murdered by the Mountain later on,) but since the prophecy stated the dragon would have three heads, he decided he would need another. And whoever this hypothetical son would be, his "would be the song of ice and fire," which is the actual name of the book series.
Still with me?
So at this point R kidnaps Lyanna Stark, who was betrothed to Robert Baratheon, and for this and other reasons, The Baratheons and Starks start a rebellion, and the Tullys and Arryns quickly join up. Until this point though, Rhaegar was basically the most beloved and respected man in all the Seven Kingdoms. That's why you saw Littlefinger pause when speaking to Sansa, the kidnapping story doesn't jive with who everyone thought R to be. Littlefinger may not know exactly the details of Lyanna's "kidnapping" and eventual death, but he is smart enough to know something doesn't fit.
So when R kidnaps Lyanna, he disappears with her for a long time, even though there is a rebellion going on. Eventually R comes out of hiding, and is killed by Robert at the Trident. Which leaves us at all the theories about Jon's parentage. But the main points are that, R was a well respected, level headed person, who supposedly did something crazy, and most readers assume he was simply attempting to fulfill a prophecy. And if you consider the Starks as ice, and the Targs as fire, you would get the literal embodiment of the Song of Ice and Fire, in Jon Snow (Targaryen.) And don't for a second think that Lyanna and R weren't married. Targaryens were known to take multiple wives, so R's marriage to Elia Martell would not have stopped him from marrying again. Jon is not a bastard, he's the son of two great houses. R would not start a war for a side chick, he needed a legitimate heir, one born through the marriage of Ice and Fire.
Rhaeger was characterized in the books as being good at everything he ever tried, including warfare. Despite being Lebron James good at warfare, Rhaegar always preferred a book and his harp to a shield and sword. Everything that Rhaegar did, including his actions with Lyanna, come across as someone performing a duty, sort of like the Spider Man mantra, with great power comes great responsibility.
If R was in fact carrying out a duty by giving birth to a child with Lyanna, he had to believe that it was for a higher purpose. R also had to know that the "kidnapping" of Lyanna would most likely lead to his death. He was smart enough to know that right off the bat, the Baratheons, Starks, Arryns, and Tullys would be gunning for him for various reasons. No matter how strong the monarchy is, you can't last if 4 of the 7 great houses are in rebellion, with the Lannisters loyalty always a dubious proposition since the Mad King had been pissing off Tywin for years at that point, you are down to just the Tyrells and Martells supporting the crown.
So R carried out the kidnapping knowing full well it would most likely lead to his death. We assume the reason was to produce a son that could fight The Others, and thus something more important than his life. R felt it was his duty, and had little choice in the matter. Elia was no longer capable of bearing children, her last child birth almost killed her, and if the "dragon needed three heads" then R would need more children. R was acting selflessly in the interest of the Realm, and paid the ultimate price for it.