Tyrion, my little ray of sunshine in a sea of misery and horror, how I love thee. Tyrion is with his father as Tywin is being notified of Jaime’s capture. Tywin is distraught at the danger the son he gives a damn about is in. Tyrion is distraught at the danger the one family member he loves and who loves him back is in. An assortment of Lannister captains and bannermen are also present, whether or not they are distraught, I couldn’t say.
I should say here that I made a mistake in my last Tyrion chapter. I usually skim military strategy bits because they don’t often make sense to me. I assumed the bit about Robb riding for Riverrun was him leaving the battle with Tywin’s forces to avoid getting captured and regroup. What actually happened (and I understand this partially thanks to this chapter, but largely thanks to the excellent TV show) was that Robb sent decoy forces to meet Tywin, who was prepared to face him, in the field and himself led the bulk of his army past Tywin and used it to attack Jaime, who wasn’t prepared for the fight.
Since the battle, Tywin has been driving his dwindling forces south to reach Riverrun in time, but, obviously, he didn’t make it. A lord allied to Lannisters by marriage bemoans Jaime’s stupidity in splitting up his forces into three, and his quickly schooled by Tywin’s brother Kevan, providing us the ignorant readers with a quick less on the geography of Riverrun: the Tully stronghold is nestled in the meeting of two rivers; when threatened, it can redirect flow to create a third side to the river triangle, turning itself into an island, so three camps are needed to successfully besiege it.
There are many details to Robb’s successful attack, but they all come down to one thing: Robb’s forces won pretty decisively. They also freed the captives, including Catelyn’s brother Edmure. Most importantly, they’re situated between Tywin and Casterly Rock, cutting Tywin off from support and positioning themselves to invade Lannister lands should they wish to do so. Tywin’s lords and captains bleat about the defeat and how they must sue for peace now.
Tyrion points out that Joffrey fucked them out of that option when he killed Ned. He smashes his wine cup for good measure as a metaphor for their dashed hopes for peace. You know things are getting serious when pottery is being smashed! There’s more bleating about trading prisoners or ransoming them, none of which the Lannisters are in any kind of bargaining position to do. Tywin gets tired of it and tells everyone, except Tyrion and Kevan, to get the fuck out.
When they’re alone, Tywin bemoans Joffrey’s folly and Tyrion softens and defends him as being only a boy. Don’t go soft on me now, Tyrion, or I’ll have to pull out the slapping gif again. (After watching the TV episode in which Ned is killed, my husband and I rewatched the slapping scene just to make ourselves feel better.) There’s more news from the south: Renly Baratheon has wed Margaery Tyrell (Loras’s sister, do you think she’s as pretty as he is?) and proclaimed himself king. Through this marriage, the Tyrells, lords of Highgarden, threw in their lot with him. In response to these news, Cersei has commanded (emphasis Tywin’s) her father to bring his forces to King’s Landing to defend it against Renly.
Joffrey doesn’t know this yet, because Cersei isn’t so deluded with maternal love to not see that her son is kind of a moron on top of his other virtues. She’s sure that should he hear, he’ll take the City Watch and lead it as an army against Renly, leaving the city open for Stannis Baratheon, Robert’s actual legal heir. Of Stannis’s plans, by the way, they know almost nothing, even Varys only hears, or says he only hears, whispers.
Tywin brings out a map and points out their rather bleak situation geographically. Let us do the same: 
Roose Bolton is north of them, Robb’s forces and allies holding the Twins. The Vale of Arryn is to the east and to the west, between them and their home, is Robb himself. Stannis’s Dragonstone is off the coast of King’s Landing. South of that, Renly’s Stormlands and the Tyrells’ Reach form about a third of Westeros’s South. South of them is Dorne, but they loathe the Lannisters for what they (specifically, G. Clegane) did to Rhaegar’s Dornish wife and her children. Not very good at making friends, the Lannisters are. Maybe a little less coin and a little more warmth, they should’ve used. Talking like Yoda, I will now stop.
Tywin wants to move to Harrenhal, unleashing G. Clegane and all of his horrors upon the countryside. He wants the lands between the Red Fork and the Gods Eye burning. He proposes to send Tyrion’s wild tribes along, but Tyrion wants to keep them for himself, he trusts them more than any forces under Tywin’s command. Tywin doesn’t care either way, as long as Tyrion can keep his wild men under control when he goes to King’s Landing.
Tyrion is unsure what he and his wild men will be doing in King’s Landing when they were just talking about Harrenhal, so Tywin connects the dots for him: Joffrey is out of hand and since Tyrion is so very good at slapping the shit out of Joffrey to the satisfaction of absolutely everyone, he’s named the Joffrey Wrangler and is being sent to take the king and his court in hand by being the Hand, or rather Acting Hand in Tywin’s place.
Verbatim, because Tyrion chapters always have funny bits:
Tywin: I am sending you to court.
Tyrion: And what am I to do there?
Tywin: Rule.
Tywin’s really not happy with all the decisions that’s been made so far, including making Janos Slynt a lord. Tywin’s a classist snob to top off his lovable personality.
Tyrion questions the assignment, knowing better than to trust his father’s apparently good will and newfound respect for his abilities. His father’s response angers him, because he realizes that Tywin’s only elevating him to his rightful position as “son” because he considers Jaime lost for good. Tyrion cares about Jaime probably more than about anyone else in the world, he doesn’t want his father giving up, he wants a fight to get his brother back, or at least a plan. None of this he says to Tywin. The Lannister family, ladies and gentleman, how much less mutual hating and fucking would they do if they just talked once in a while?
As one last thing, Tywin orders Tyrion not to take Shae (“the whore”) to court with him. Now that’s just spite. The court is filled with prostitutes. Littlefinger runs brothels! So, let’s say he doesn’t take her, it’s not as if he won’t have women in King’s Landing. It’s all a moot point, really, since the first thing Tyrion does upon getting back to his room is to tell Shae to pack her bags. They’re going to the capital. I hope he slaps Joffrey the moment he gets there.