Eddard Stark is the lord of the north, but he's friends with (and less powerful than) King Robert Baratheon. So when Robert asks Eddard to be the Hand of the King (which is like being Prime Minister, but less fun), Eddard can't really refuse. (What makes this even less fun is that the old Hand died under mysterious circumstances; and he was also a good friend.) So Eddard decides to go to the capital, King's Landing. There's just one problem: the Stark family symbol is a direwolf and all six of Eddard's kids have one of these giant wolves – and they're not really city pets.
Unfortunately, the Stark family doesn't get along so well with the Lannister family. This is an issue because Cersei Lannister is the queen; and her twin brother Jaime is a powerful knight. But the third brother is Tyrion, and he seems like an okay guy: he's funny and smart (and a dwarf, by the way). By the way, the Lannister family has a secret: Cersei and Jaime are incestuous. This is so dangerous a secret that Jaime throws young Bran Stark from a tower when he sees the Lannister twins in the act.
Down in King's Landing, Eddard gets caught up in the court politics and conspiracy. Eddard doesn't know whom to trust: should he trust his wife's old friend, Petyr Baelish, who is now a powerful politician (read: schemer)? Should he trust the old wise man, Maester Pycelle, who seems to be an ally of the queen? Or the spymaster, Varys?
After another assassination attempt on Bran's life, Catelyn comes down to King's Landing with the knife that was used to try to kill Bran. (Luckily, Bran has a wolf to protect him.) Petyr tells her that the knife is Tyrion's. So when Catelyn heads home and runs into Tyrion (coming back from the Wall), she decides to take him prisoner and takes him to see her completely crazy sister. That doesn't go well and Tyrion wins his freedom and a small army.
Meanwhile, Eddard learns that King Robert has a lot of illegitimate children and they all have his dark hair, which is very different from Robert's children with Cersei. In fact, Cersei's three kids look a lot like… Jaime. Yeah. So now Eddard knows the big secret, which isn't just gross, but is also a political issue: Cersei's kids aren't true heirs to the throne. Eddard tells Cersei that he knows about her incestuous relationship and that she should get out of town. She responds (we're paraphrasing here): "You gonna make me? You and what army?"
When Robert dies (under mysterious circumstances), Cersei arrests Eddard. She promises to protect his two daughters Sansa and Arya (the two he brought down to King's Landing) if, and only if, he'll confess to treason. He does just this, and Cersei's son Joffrey – now the king – has Eddard killed.
So now we have war in Westeros: Eddard's oldest (trueborn) son, Robb Stark, is now in charge of the north and he has an army. Meanwhile, the Lannister army has been attacking the Tully family in the Riverlands. (It doesn't help that Catelyn is a Tully by birth.) Although he's very young, Robb turns out to be a pretty good commander, even capturing Jaime Lannister. But Robb isn't sure who should be king of all Westeros, until his followers tell him that he should secede and be the king in the north, like in the old days. It's funny to see that even in fantasy worlds, people still love them some nostalgia.
Here there be… dragons?
Before Robert Baratheon was king of the Seven Kingdoms, they were ruled by the Targaryen family, who had the charming habit of marrying brother to sister to keep their blood pure. As with most people who are interested in "blood purity," some of the Targaryens were bughouse mad, which is part of the reason why they got overthrown (by Robert, Eddard and others). Now all the Targaryens are dead except for a young exiled prince – Viserys, who really wants to get his throne back (and is a jerk) – and a younger exiled princess – Daenerys, who just wants to make her brother happy. Although the Targaryens once had dragons (that's how they conquered Westeros), Viserys and Daenerys have nothing: they have to rely on the kindness of people. (And people aren't really that kind in a George R.R. Martin book.)
While they're in exile, Viserys marries (or sells) his sister to Khal Drogo, a powerful leader of a barbarian tribe. (These horse-riding barbarians are called the Dothraki.) As part of her wedding present, Daenerys is given three fossilized dragon eggs (and a bunch of other stuff, but keep your eyes on those dragon eggs). Khal Drogo is strange and scary, at first, but Daenerys adapts to the Dothraki way of life and is happy. Also, she gets pregnant. Meanwhile, Viserys remains focused only on his own feelings and continues to be a jerk. Viserys even threatens Daenerys and her unborn son, but Khal Drogo won't have that and kills Viserys in a way that is both poetic and gruesome. It's really a win-win: everyone loves that scene. (Except maybe Viserys.)
After Daenerys survives an assassination attempt ordered by King Robert, Drogo decides that he needs to kill his wife's enemies (aw, what a sweet anniversary present). But in the process, Drogo gets wounded (less romantic). Daenerys asks the help of a wise woman named Mirri Maz Duur to try to heal Drogo, but Mirri betrays Daenerys' trust and kills both Drogo and her unborn son. In response, Daenerys grieves by burning Mirri alive on a pyre. Take that. This awesome revenge has the added benefit of hatching those dragon eggs, which it turns out weren't fossilized at all. Who would've guessed that a fantasy novel would have dragons?
It's hard out there for an illegitimate son
Just so we're clear, in Game of Thrones, George R.R. Martin uses the word "bastard" to describe an illegitimate child (someone whose parents weren't married). Since that's got some negative connotations these days, we'll just call Jon an illegitimate son.
Jon Snow is Eddard Stark's, yes, illegitimate son, and while Eddard loves him, Catelyn kind of hates him. So when Eddard goes down south to King's Landing, Jon can't stay home. Instead, he joins the military order of the Night's Watch, a bunch of poor guys who guard the giant Wall in the north. The Night's Watch used to be a well-respected institution, but now most of them are convicts who were given the choice between losing limbs/dying or joining the gang.
Unfortunately for Jon, his uncle Benjen Stark goes missing in the north; also, Jon is terrible at making friends because he's kind of a stuck-up bully. (And to think, he's one of our favorite characters.) But Jon stops being such a jerk and makes some friends, including fat, cowardly Samwell Tarly. On the other hand, Jon also makes one enemy out of their fighting teacher, Alliser Thorne, so he needs all the friends he can get.
That's especially true when Jon sees firsthand that there are monsters up in the north: two of the Night's Watch were killed and turned into some sort of snow zombie and Jon barely manages to kill one of them. So even though there's a lot of crazy civil war stuff going on in the south, Jon's friends make him see that the real monsters are in the north with them.