Robb is the oldest son of Eddard and Catelyn Stark; he's about the same age as Jon Snow and older than Sansa, Arya, Bran, and Rickon. Robb is also good friends with his father's ward, Theon Greyjoy. Robb is left in charge of Winterfell and he calls up his army after Eddard is captured. Though he's very young, Robb turns out to be pretty good at this war thing, even capturing Jaime Lannister. At the end of the book, he is declared king in the north by his men. Score.
Robb Stark is a hero in the making: when we meet him he's only (reality check!) fourteen years old (2 Bran 1.15), but he's already practicing to be the lord of Winterfell. That's because he's the oldest son of Eddard, so he's destined to be the lord of the north. Unfortunately, that happens much sooner than anyone expected, what with that little "sorry we beheaded your father" issue in the south.
In a way, what happens to Robb might be like what happened to his father: Eddard Stark was the second son, so he wasn't supposed to be the lord of Winterfell; but after Eddard's father and older brother died, Eddard inherited the position for which he (maybe) wasn't ready. In just this way, Robb probably looked forward to having several years to learn to be a lord. But since his father and mother left – and especially after Eddard's death – Robb has to grow up very fast. (See "Coming of Age" for more on that topic.)
However, although Robb is way young for this position, he seems to be doing pretty well. When Catelyn Stark is too sad about Bran to make decisions, Robb steps up (15 Catelyn 3.12). When Robb calls the Stark bannermen (the other noble houses sworn to House Stark), he stands up to the people who think he's too young to lead; and when the Greatjon Umber threatens violence, Robb (with the help of his wolf Grey Wind) stands up to him and coolly jokes about it (54 Bran 6.35). And (just one more example, we promise) Robb comes up with a plan of battle that not even Tywin Lannister could anticipate (56 Catelyn 8.71). This is all Robb the Lord, the Young Wolf.
All this heroism can make us forget that Robb is still a young guy, so our author sticks in a few young-guy moments to help us remember. When Robb talks to his mom about how she's obsessing over comatose Bran, he nearly cries and bites his lips like he did as a child (15 Catelyn 3.23). Although Robb stands up to his bannermen, he confesses to Bran later that night that he was terrified when Greatjon almost attacked him (54 Bran 6.40). And even though Robb comes up with this great plan of attack, there are a few holes in the plan that his mother helps him to fix. (Wouldn't it be great if all generals had help from their mothers?)
So, Robb might be the new lord of Winterfell, and he might be doing a real good job of it, but he's definitely not perfect.
Robb's best friend (non-family category) is probably Theon Greyjoy, a nineteen-year-old rascal who kicks decapitated heads and talks about his sexual conquests (2 Bran 1.11, 38 Bran 5.10). While Robb might enjoy his company, Jon Snow and Bran don't like him all that much (2 Bran 1.12, 38 Bran 5.12). We're not sure whom to trust on this one.
Theon does seem a little racsally; for instance, he takes a chancy shot with a bow-and-arrow, which happens to save Bran's life, but could have possibly killed him (38 Bran 5). Still, when he pledges to fight for the Starks and says that "Lord Eddard is a second father to me," we want to believe him (15 Catelyn 3.89).
Eddard may be fatherly to Theon (Eddard is fatherly to everything – he probably teaches important fatherly lessons to streams and rocks on his property), but Theon isn't Eddard's son. Theon is Eddard's ward and hostage. In fact, Eddard keeps Theon around to make sure that Theon's real father doesn't pull any more crazy stunts like declaring independence. So maybe we should keep that in mind: he's not just a family friend, he's also a potential enemy. Maybe that's why we feel so uncertain about him.