SPOILERS AHEAD, OBVIOUSLY.
Rebecca already touched on this theory in her roundup of Game of Thrones book reader theories, however this video helpfully breaks down what is probably the most famous theory for non-book reader dum dums like myself. Basically, the long and short of it is that Jon Snow wasn’t Ned Stark’s son at all, but actually the spawn of Lyanna Stark, who had either been kidnapped or run away with Rhaegar Targaryen (son of the Mad King and brother to Daenerys and Vicerys) and had a secret child with him which she made Ned swear to take care of shortly before she died. Lyanna, of course, had been betrothed to Robert Baratheon, which is what set the entire chain of events in motion leading to Baratheon dethroning the Mad King and taking over King’s Landing.
OK, so but what does this mean? Well, being the son of a Stark and a Targaryen would make Jon “the son of ice and fire personified,” which suggests that Jon is very important — possibly possessing both the Stark superpower of warging (the ability to enter the minds of animals) and Targaryen powers having to do with dragons and fire. (Warging dragons??) It also would mean that Jon Snow is possibly the rightful heir to the Iron Throne.
So what does this all mean? Hell, if I know. If the theory is true, then the secret died with both Ned and Lyanna, which means that Jon Snow isn’t likely going to quit moping around anytime soon. Brings all new meaning to “You know nothing, Jon Snow.”