Bran Stark is the fourth child of Eddard and Catelyn Stark; brother to Robb, Sansa, Arya, and Rickon; and half-brother to Jon Snow. Bran falls (read: is thrown) from a tower and breaks his back, and he spends the rest of the book talking about the world. (Wow, when we put it that way, Bran sounds kind of boring; and yet we love his chapters because they tell us so much about this strange and exciting world.) He spends a lot of time with Osha, Old Nan, and Hodor.
Let's get this straight: Bran may be a mystical child, having coma-dreams that turn out to be real visions of what's going on all over the world (see 18 Bran 3), but he's also a seven-year-old who is going to be crippled for the rest of his life. (And that's your reality check for Bran: he's seven when this book starts and when he sees his first execution [2 Bran 1.1]. Yikes.)
We get to see how a whole bunch of characters react to Bran's fall: Eddard is sad but carries on (just like he probably did when his dad, brother, and sister were killed by the Targaryens), but Catelyn is crushed by the love she has for her kids (which is a large part of her character). Cruel Joffrey doesn't care, and Tyrion is surprisingly sensitive.
But most of all, we get to see how Bran tries to cope with this: how he dreams about overcoming his disability and becoming a knight; and how he deals with his bitter frustration of not making his dreams a reality.
In some ways – and feel free to disagree with us on this – Bran doesn't seem to affect the plot much in this book. That is, he thinks and feels a good amount, but Bran doesn't do a lot to change the plot here.
But maybe that's what he's here for: his chapters help explain this world to us and show us other sides of the other characters. (For instance, Robb cries in a Bran chapter [25 Bran 4], not in an Eddard or Catelyn chapter.) While Eddard, Catelyn, Robb, Sansa, and Arya are dealing with the difficulties of politics and a looming civil war in the south, Bran is the only Stark who has any idea that there might be something else out there. Something dangerous and cold. And maybe that's why Bran names his wolf Summer; because while all his brothers are fighting the Lannisters, Bran is getting ready to fight the real enemy: winter.
Rickon is the youngest of the Stark children. Rickon is three years old when the book begins (2 Bran 1.70), which leads to a very telling comment, when Robb complains about Rickon: "He can't be a baby forever. He's a Stark, and near four" (54 Bran 6.87). (What? He's four already and he hasn't yet witnessed an execution? What a baby!) This is about the clearest way that anyone can say that the Starks have to grow up fast.
But Rickon is still a baby. He names his direwolf Shaggydog, which is a pretty babyish name for a wolf. In fact, Rickon doesn't do a lot in this book except act childishly. And yet, when he gets upset that everyone is leaving, we don't get super annoyed with him because it's really sad: first, his older brother goes into a coma, then his half-brother leaves for the Wall, then his dad and sisters leave, then his mom leaves, then his oldest brother leaves to fight a war, then his father dies… Really, let's all cut Rickon some slack.
Actually, we can't exactly do that. Maybe that's why he's here in the book: to remind us that while we might want to protect these children, they're going to have to grow up quickly to face what's coming.
Osha is a great example of how working hard can be rewarded: she starts as a prisoner to the Starks after she attacks Bran (38 Bran 5); becomes a servant in the kitchen (54 Bran 6); and eventually becomes trusted enough to literally carry Bran (67 Bran 7). Now that's what we call upward mobility.
But Osha is really important to the book because she was originally a wildling who lived north of the wall. While all the Starks and their servants are focused on the Lannisters (or being killed by the Lannisters – it's hard being a Stark guardsman), Osha and Bran are the only two characters who have some idea that something even worse is about to come from the north. For an example of this, check out her talk with Maester Luwin about the history of Westeros: they both know some history, but Osha believes in the old tales (67 Bran 7).
And that brings us to Old Nan, the source of all the old tales in this book. She's a medieval version of Wikipedia, including the whole factual questionability thing. Old Nan is so old that no one knows how old she is, including her (25 Bran 4.8). She was originally brought to Winterfell as a wet nurse (a woman who breastfeeds someone else's baby), but she stayed on as a servant.
Old Nan is really the only entertainment they have in Winterfell: that's what life in an oral culture is like. (Well, there are books in Westeros, but a lot of information is oral; we don't see Bran reading under his bed sheets at night with a light.) Old Nan is very useful in the book because she tells Bran (and us) lots of old histories, including stories of the Others. For instance, check out that story she tells in 25 Bran 4, where the Others have hunting spiders and corner the hero. Creepy.
Hodor is the simpleminded giant who works in the stables (25 Bran 4.8) and only says "Hodor." (So everyone calls him "Hodor," even though his actual name is Walder.) He's very useful at carrying Bran around (which is kind of funny: he goes from being a stable boy to acting as Bran's horse for a while). He's somehow related to Old Nan, but since no one knows how old she is, no one knows how they're related. If Bran is a smart young boy with a damaged body, then Hodor is like the opposite: a giant body without much of a brain. They make a good team. (Hey, that's kind of like Tyrion and Bronn.)