Tyrion Lannister is the misshapen dwarf son of Tywin Lannister and the younger brother to Cersei and Jaime. Tyrion has quite an adventure: he visits the Night's Watch, gets abducted by Catelyn Stark, escapes (with the help of Bronn) and recruits the Mountain Clans, and finally fights in his father's army. It's a pretty full day for this guy.
Tyrion is like a stereotypical hacker/nerd character from 90s movies: he's pretty weak physically and not very attractive (except, instead of just being pale, like a hacker, Tyrion is a dwarf with mismatched eyes); but at the same time, he's super powerful because of how smart he is: he can smart himself out of pretty much any bad situation. Pretty awesome.
Here's why we love him: (1) All Shmoopers identify with the smart guy…right? (2) In a world full of people beating each other up, the dwarf kind of seems like an underdog. (3) Tyrion loves reading. If you're in the middle of an 800-page book and enjoying it, chances are you're going to sympathize with the character who also loves to read. As Tyrion explains to Jon, "My mind is my weapon […] and a mind needs books as a sword needs a whetstone, if it is to keep its edge" (14 Tyrion 2.35).
Basically, Tyrion has lots of positive nerdy qualities: he's smart and funny, and he's also absurdly rich and from a powerful family. He's the medieval fantasy version of Bill Gates (minus the philanthropy, plus a love of prostitutes).
What we love even more about Tyrion is that he isn't a dream character for the readers: who would want to be Tyrion? Sure, he's got good qualities and some lucky strokes, but being Tyrion isn't all that great. As he explains to Jon Snow, being a dwarf is sort of like being a "bastard" (6 Jon 1.76) or even like a cripple (which is why he can figure out a saddle for crippled Bran to ride – his saddle is already like it).
But our friend Tyrion is okay with the hand life dealt him (like being a misshapen dwarf with mismatched eyes and a family that mostly hates him). For instance, Tyrion knows that he'll never be a great knight like his brother Jaime. Check out the scene when Tyrion is on the High Road with Catelyn Stark's party; all of the other fighters are screaming out their battle cries: "Tyrion felt a sudden urge to leap up, brandish his axe, and boom out, 'Casterly Rock!' but the insanity passed quickly and he crouched down lower" (32 Tyrion 4.82).
Whatever else he is, Tyrion is a survivor who knows that he's not going to survive by pretending to be a brave warrior; and he doesn't pity himself over it. (Imagine Viserys, the poster-boy of self-pity, in that position.) Instead, Tyrion realizes his limitations and still works toward winning by emphasizing his strengths.
Tyrion is also really important in this book because he gives us our only non-Stark POV on what's going on in Westeros. So we don't just see the Lannister-Stark war through Stark eyes; through Tyrion, we get a view of what the Lannisters are doing and thinking, too.
He also complicates what might otherwise be a simple good/evil situation. We might root for the Stark army (because the Starks are so nice, even though they have major flaws), but we also root for Tyrion. And the Starks and the Lannisters can't both win. So by making us love Tyrion, George R.R. Martin teaches us that everything won't be great in the end: someone we like might lose – or die.
(Martin emphasizes this problem by making Jon Snow and Tyrion into friends. Two of our favorite characters like each other, but they're playing for opposite teams. Does Hallmark make an "Oh, hey dude, sorry my nephew cut off your dad's head" card?)
Bronn is originally a mercenary for Catelyn Stark (recruited in the Old Crossroads Inn, when Catelyn captures Tyrion). But Bronn becomes closer to Tyrion (they both have a dark sense of humor) and eventually switches to Tyrion's side, because that's where the money is.
Bronn fights for Tyrion when he needs a champion in the Aerie's trial by combat (41 Catelyn 7). Like some of the Night's Watch, Bronn is a reminder that "noble family" doesn't equal "good at fighting." In that way, he's the perfect companion for Tyrion: Tyrion has the brains and the money; Bronn has the… brawn. But even though Bronn is a mercenary for Tyrion, they seem to get along pretty well.
Oof. For a character who isn't actually in this book, Tysha sure brings up a lot of feelings. She's the common girl who marries Tyrion and teaches him how to love. (Aww.) But then she's revealed to be a prostitute and someone who never loved Tyrion at all. (Boo.) She never shows up in this book, but her story sets up a lot of important information about Tyrion, including the fact that he's capable of love; and that his relationship with his father is very, well, complex. In a way, her story also foreshadows the story of Shae.
Shae is Tyrion's lover after he makes his way to his father's army. She's funny and smart, and a pretty good match for our guy. One problem: Dad doesn't approve. This is a dilemma that Tyrion faces at the end of the book: his father is finally showing some approval of him (by sending him to be Hand of the King for King Joffrey), but he demands that Tyrion leave Shae behind. Will he?
Podrick has a very small role in this book as Tyrion's squire (the guy who helps a knight get into armor) because Tyrion's major fights are all off of the battlefield.
Jyck and Morrec are Tyrion's servants for the first part of the book, until they are killed by mountain clansmen on the trip to the Aerie, isolating Tyrion even more.