Jon Snow kind of gets the short end of the stick in this book. He is Eddard Stark's illegitimate son, and we may never know who his mother was. (And let's be honest, some people doubt that Eddard is even the father.) Jon was brought up with his noble half-siblings, but when Eddard goes to the south, Jon joins the Night's Watch in the north and has to adapt to a whole new social scene.
Jon Snow is sort of a cool-dude loner character – like Wolverine from X-Men – except he has a bunch of packs to choose from. In fact, Jon's biggest problem is that he can't belong to multiple packs at the same time. One the one hand, it's kind of an embarrassment of riches. But on the other, more important hand, he's kind of an outsider. He doesn't fit in with his family because he's not a "trueborn" son, but he doesn't fit in with the Night's Watch gang because he's the son of a lord. Lose-lose. At the end, he wants to be a Stark and help his brother Robb avenge their father; but he also wants to keep his vows to the Night's Watch. And he can't do both at the same time.
Somehow, Jon is still loyal, friendly, and generous to everyone around him. First, he gives his little sister Arya her sword (11 Jon 2): everyone else wants Arya to be a little lady, but Jon looks at the situation, sees his sister is a tomboy, and basically says, "I'm going to support you, even if no one else will." And check out his friendship with Samwell Tarly: Sam is a self-described coward, but for the most part, Jon accepts him for who he is. And the list goes on.
So as much as he's on the outside, he manages to form meaningful relationships on the inside, too. This double perspective allows him to see things that other people maybe can't see; like, say, that Joffrey is a "little shit" (8 Arya 1.63).
Jon Snow, Hero
In some ways, Jon Snow is like the hero of your awesomely average young adult novel: he's a bit of an outsider, but he's super smart and very skilled at fighting. (Think Katniss from The Hunger Games.) Jon doesn't know what his place should be in the world: he's an illegitimate child, so he can't inherit anything from his dad (Eddard Stark), but at the same time, he's been raised in this world of noble families. And thanks to all that noble attention, Jon Snow is pretty good at the noble pursuits, like sword-fighting and riding. A bit of a misfit, a bit of a skilled fighter, and pretty smart, this guy certainly fits the bill for hero.
Jon Snow, Jerk?
There are a few things that keep Jon from being a classic hero: he's kind of a bully and a stuck-up jerk (as Donal Noye points out to him [20 Jon 3]). But Jon makes up for all of his bad moments (and there are quite a few in this book) by learning from his mistakes.
And here would be a good time to stick in our reality check: At the start of the book, Jon is only fourteen years old (2 Bran 1.12). So, yeah, sometimes he messes up, but he's young, so we have to cut him some slack. He'll learn.