Tywin Lannister is the patriarch of the Lannister family and the Lord of Casterly Rock. We've only met this guy in a few chapters of both A Game of Thrones and A Clash of Kings, but he's made his presence known all the same.
For most of this volume, Tywin spends his time strategizing his war against Robb Stark and the North. There are no point of view characters present during these times, so we're left with hearsay from other characters—but that hearsay says Tywin Lannister can fight a war.
As Brynden Tully tells Catelyn: "And Tywin Lannister is no man's fool. He sits safe behind the walls of Harrenhal, feeding his host on our harvest and burning what he does not take. Gregor is not the only dog he's loosed" (8.Catelyn.105). Couple things here. Brynden seems pretty savvy when it comes to strategy, so we're going to take his word that Tywin isn't a fool. And also, the idea that Tywin can set Gregor Clegane, the Mountain, loose implies that Tywin has power over that bruiser of a man in the first place. That's like calling Heracles your errand boy.
Tywin also demonstrates how truly powerful a figure he is on the Seven Kingdoms's scene. This is probably best shown when a Florent proclaims the Lord of Light favors Stannis in the throne room. Joffrey gets belligerent and injures himself on the Iron Throne, but Tywin just "raise[s] a finger" (66.Sansa.47) and Ser Meryn murders the Florent on the spot. Like a boss.
This powerful influence extends toward his children, though, and it is pretty evident Tywin won't win any father of the year awards. When Tyrion lies severely injured in some unknown place, listening to the groans and whimpers of dying men around him, he wonders if he is crying and immediately thinks "[h]e must not let his father see" (68.Tyrion.1). Father of the year awards? Heck, a mug reading "#1 Dad" would probably crack in his hands.