While an antagonist in the series, he's one of my favorite bad guys in fiction. Before the show begins, he's the Warden of the West, and Lord of Casterly Rock. In his youth, he was good friends with King Aerys II and he made him his Hand of the King. After fearing his power, the Mad King caused Tywin to resign (especially after Cersei was turned down as a bride to Rheagar) and he eventually is the reason the Mad King dies when Tywin uses his army to take over King's Landing. He spends most of the time in peace until Ned Stark challenges him. But it was Robb Stark that would concern him as he proved to be too strong to master in battle.
While named Hand of the King for King Joffrey, Tywin gives the role to Tyrion until he was done at war. While traveling around, he accidentally finds Arya Stark (whom he thinks is a poor Northern girl) and makes her his cupbearer. I thought it was cute that he mentored her for the most part, Arya probably got the most respect out of all the characters Tywin had came across.
While he never could best Robb Stark in battle (though they never directly fought one another) he was always a powerful battle commander. He uses his general-like role to defeat Stannis Bartheon at King's Landing after Tyrion gives his father enough time to make it back home.
After Stannis is defeated, he becomes Hand of the King again, for real. He doesn't go back to battle, especially after organizing Robb Stark's death at the Red Wedding. While Cersei, Joffrey, and Tyrion all were strong, confident, and powerful people, they all fell under Tywin's shadow. Even the evil boy-king is terrified of what his double-grandfather can really do with all his Lannister gold and power.
When Joffrey dies, he takes pride in having Tyrion as the suspect and arranges a kangaroo court in order to press Tyrion to join the Night's Watch. Tyrion shocks everyone by demanding a trial by combat.
I don't think Tywin liked how Oberyn Martell was winning against the Mountain for the trail by combat. You can see the deep look of concern on his face when Oberyn tries to get Ser Gregor Clegane to admit that Tywin ordered him to murder Princess Elia Martell and her royal children. He seems pretty happy to condemn Tyrion to death.
Though he ironically dies when Tyrion escapes and shocks his father while he was on the royal toilet. He takes Shae and has an affair with her, which I found a little odd considering he would mock Tyrion for his love of "lower women". You could imagine that it was done to shame Tyrion but how likely was Tyrion would know about it?