Character Analysis
Tywin
Tywin is the father of Jaime, Cersei, and Tyrion Lannister. He's the head of House Lannister and he's a very hard man (57 Tyrion 7.31). He's noted as being dangerous, violent, and willing to do anything to win. For instance, when the Lannisters join in with Robert's rebellion, Tywin's army raids King's Landing and kills all of the Targaryens they find, including the children. Was that an accident or did Tywin order it?
In addition to being dangerous to his enemies, Tywin Lannister doesn't show any particular love toward his children. In fact, he hardly shows any emotions at all. As Tyrion notes, "Lord Tywin Lannister did not smile. Lord Tywin never smiled…" (57 Tyrion 7.76). Now, Tywin does show a little bit of pleasure on his face, but that has to do with Tywin's plan to destroy the Stark army. Because that's the kind of guy he is: amused by the thought of violence. (Oh, how upset he is when Robb Stark outsmarts him.)
So Tywin might be the exact opposite of Eddard Stark: not a great dad, but a very dangerous manipulator and schemer.
Ser Kevan
Kevan is Tywin's only surviving brother and also his right-hand-man. Or maybe he's more like Tywin's yes-man. That is, as Tyrion notes (and notice how we always have to go to Tyrion for these impressions of the Lannisters), "Ser Kevan seldom 'had a thought' that Lord Tywin had not had first" (63 Tyrion 8.17). Ha!
Kevan doesn't do much in this book, but he shows us that Tywin Lannister isn't particularly affectionate with anyone in his family. Also, by being a yes-man, Kevan makes Tyrion seem smarter and stronger.
Tywin's Army
Tywin Lannister's army includes many nobles that are sworn to House Lannister; they pass through his book pretty quickly, but in case you get confused, here's a list of nobles that make it through the fight with Robb Stark's decoy army:
Flement Brax
Addam Marbrand
Harys Swift
Leo Lefford
But check this out: while we know only a few names of the people who survive the battle, we know many of the names for people who are captured by Robb alongside Jaime Lannister:
Gawen Westerling
Quenten Banefort
Regenard Estren
Willem Lannister (Kevan's younger son)
Garth Greenfield
Tytos Brax
Mallor the Dornishman (a mercenary)
Cleos Frey and Tion Frey (whose mother is a Lannister, which is why they're fighting with the Lannisters rather than with the Freys)
So, really, we see more people get captured than we see fight successfully. No wonder the end of the book sees the Lannister army needing to retreat to a safe place.