Robert Baratheon is king of the Seven Kingdoms after the Targaryens are kicked out (which Robert has a little something to do with). He was raised with Eddard Stark (though they haven't seen each other in years) and loved Eddard's sister Lyanna. But since Lyanna died, Robert was advised to marry Cersei Lannister, which was not the best move. Robert has two younger brothers, Stannis (who doesn't appear in this book) and Renly. Robert spends most of his time as king eating, drinking, hunting, and sleeping around. And because of that last one, he's got a bunch of kids wandering around this book. In the end, Robert is killed while hunting drunk (HUI: Hunting Under the Influence).
We feel kind of bad for Robert Baratheon, which is unusual because he's a stinkin' king. For one thing, notice how far down he is on the list of important characters. He's the freakin' king of the Seven Kingdoms and he's still less important than a seven-year-old who gets his legs broken.
But the real reason we feel bad for King Robert is that he discovers one of those universal truths that always bite us in the rear: sometimes, it's more fun to go for something than it is to actually have that thing. Here's Robert's diagnosis of his problem: "I was never so alive as when I was winning this throne, or so dead as now that I've won it" (31 Eddard 7.46). And on top of that, he lost the woman he truly loved (Lyanna Stark). So it's no wonder that he got fat and spends all his time drunk or having sex with random women.
That said, let's be honest about what sort of a king Robert is: not a very good one. We may be able to understand Robert's unhappiness: who would be happy in that situation? Do you think Eddard would be happy if he lost Catelyn and had to spend all his time with Petyr and Varys? But, still, we have to hold him accountable for the screw-ups of his reign, including getting into debt and giving more power to the power-hungry Lannisters.
He'd probably be a great friend, but he's a pretty terrible king. Still, he's a better at it than bloodthirsty Joffrey; and he's willing to face the hard truth and acknowledge that his "son" is kind of a bad seed (31 Eddard 7.48). (If only he followed that thought a little further instead of just getting drunk.)
Renly is the youngest Baratheon (he's twenty-two) and he looks like Robert back when Robert looked awesome (young, strong, not yet crushed by life). Renly serves as master of laws on Robert's council.
But he doesn't seem very interested in the law when he tries to get Eddard to take over the castle after Robert's death; or when he leaves King's Landing and declares himself the rightful king (even though his older brother Stannis should be king). At those times, Renly seems like just another scheming nobleman (like Tywin Lannister or Petyr Baelish). So maybe he's not all that much like Robert, who couldn't scheme his way out of a paper bag.
Renly might just be around to help us see who Robert is now: fat and maybe a little naïve.
A large part of A Game of Thrones is Eddard's investigation into Robert's illegitimate children, which helps him figure out that Cersei's kids aren't Robert's. So Eddard scours King's Landing and finds black-haired Gendry (the apprentice armorer under Tobho Mott) and black-haired baby Barra. These characters don't do much in this book except solve the mystery of the blond Baratheons (the solution: they're not Baratheons, just Lannisters). This isn't just gross (incest, remember?), but also a political matter: if Joffrey isn't the king's kid, then he shouldn't be able to become king.