Jon Snow’s origin takes us back to the time when Rhaegar Targaryen was alleged to abduct Lyanna Stark of Winterfell. This sham led the outburst of Robert’s Rebellion. Rhaegar eventually returned to the fight only to be defeated and killed by Robert Baratheon. Lord Eddard Stark, a friend to Robert Baratheon rode to war alongside him to avenge his father and brother and to rescue his sister. With the death of Rhaegar, Robert sealed the victory for the rebellions. After taking King’s Landing; Ned continued his search for his sister Lyanna.
Rhaegar left Lyanna in the tower of Joy. She was guarded by Lord Commander Gerold Hightower and Ser Arthur Dayne (both) of the kingsguard. They welcomed Ned Stark and his companions with a fierce melee. In the end both of the Targaryen Kingsguard were killed as well as most of the combatants on Ned’s side. Ned himself was saved by Howland Reed.
After defeating the Targaryen Kingsguard, Ned stormed into the tower and found Lyanna dying from childbirth. She just gave birth to Rhaegar’s own son. With a desperate attempt to save her newborn’s life Lyanna, while fading into death, pleaded with Ned to promise her that he would keep her son safe. She also pleaded to keep the newborn’s true heritage a secret from Robert. To keep the promise he made to his dying sister Ned Stark resolved to pass Lyanna’s son off as his own bastard son therefore raised him in his own home bringing a great blow to his honor. Ned also decided not to use the actual name of the child; the one which was revealed by Lyanna in her final moments. He decided so because of the Targaryenish heritage and style of the name. It would have given away the child’s real parentage. So he named the boy after his great friend and mentor Jon Arryn, Lord of the vale and according to the Westeroshi custom the boy grew up with the surname ‘Snow’, as all bastards of the North are acknowledged in such a way. At the end of season 7 we get confirmed that Jon was not actually a bastard at all. Rhaegar apparently annulled his marriage to Elia and married Lyanna.
Jon Snow spent the next seventeen years in Winterfell raised as Lord Eddard Stark’s bastard son, alongside his trueborn children and his wife. Despite sharing a happy marriage, Jon’s presence at Winterfell would serve as a constant source of friction between Ned and Catelyn. As Jon was never legitimimzed, Catelyn was never his ‘stepmother’ and so did nat have any obligations to him whatsoever. She never mistreated Jon but definitely was cold towards him, viewing him only as a living remimnder of the time that Ned dishonored her. Catelyn later confesses to her daughter-in-law Talisa about a time when she cursed Jon and he caught the pox. Out of guilt for praying for is death, she stayed with him through the night and prayed to the gods to let him live, accepting the fact that the boy was not to blame for his husband’s sin. Jon lived; but her guilt and acceptance was not enough to make her love Jon as her own, something that she eventually regrets about. (Read about the conversation here.)
Understably, Ned never told anyone about the true heritage of Jon Snow; even to Jon himself. When pushed by the king Robert Baratheon about who Jon’s mother was Ned couldn’t deny an answer and went as far to concort a vageue explanation that Jon’s mother was a lowborn woman named Wylla that he met during the war.
Jon grew up feeling like an outsider at Winterfell due to his illegitimate status. Despite of his attendance in Winterfell’s formal occassions being restricted and he not having the permission to sit inside the Lord’s table with his family, it cannot be said that Lord Stark or Lady Catelyn mis-treated Jon. Rather he was well off than the other bastards and Ned would see that he was treated well. He was loved by his father, most if his siblings, was never hungry or poor, lived in his father’s castle and had a noble’s upbringing.
Jon Snow is a character of clear moral compass and a true sense of honor. He tries his hardest to abide by these moral codes when he must take a difficult decision. He has always admired and idolized his father. Ned also had a fairly warm corner for Jon but as a result of Catelyn Stark’s cold disdain he was careful not to give him more attention than his lawful children by Catelyn. This whole complexity reflects in the scene where Ned and Jon part ways. “You are a Stark. You may not have my name, but you have my blood.”
His relationships with half-siblings were more friendly particularly with Robb and Arya. Robb and Jon were of the same age therefore became playmates and comrades in training. On the other hand Arya due to her tomboyish personality was also something of a social misfit like Jon and this common ground helped them to bond over. Bran and Rickon may not be as close to Jon as the previous two but they also have well-wishes and love for Jon and vice-versa. At the beginning the one sibling of Jon who took after Lady Catelyn’s view towards Jon was Sansa. She was the only one who was also cold towards Jon. But later as she experiences the bitterness of the outer world Sansa learns to value her family and both Jon and Sansa share a heart-warming relation with each other.
Outside the family, Jon develops an interesting relation with Tyrion Lannister. Tyrion also being a social outcast finds an interest on Jon. Jon’s personality somehow impresses him and so Tyrion builds up a friendly approach to Jon. Their friendship has a small but effective development while they share the road to The Wall and durig Tyrions stay at the wall. Though Starks and Lannisters share nothing but cold hatred, these two outcasts of the two families bond over their shared ground. Later they re-unite after a long time and continue to cooperate each other.
Jon Snow also creates a similar type of impression on the Lord Commander Jeor Mormont. So much that the lord commander takes the task of training Jon to be the next Lord Commander. His helping nature and headstrong approach makes him famous with the underdogs at night’s watch. Outcasts such as Samwell Tarly, Grenn and Pyp associates themeselves with Jon literally to the end of their lives. Jon being compassionate and honorable is very protective of his fellow outcasts.