Tywin Lannister was the Lord of Casterly Rock until 300 AC. He also held the titles Shield of Lannisport and Warden of the West. Up until his death, Tywin was one of the most powerful lords in Westeros, and was the father of Jaime, Cersei, and Tyrion Lannister. He apparently despised Tyrion, partly for his deformity in being a dwarf, and killing the love of his life, Joanna, during childbirth. As of the end of the War of the Five Kings his branch which once ruled Casterly Rock was exterminated, with his brother, Kevan Lannister having continued the family line.
In the early years of being Lord of Casterly Rock, Tywin focused on restoring the image of House Lannister whose reputation was low thanks to the actions of his father, Tytos Lannister. Tywin's reformative years of the Westerlands were effective with the peak of it being the extermination of rebel houses Tarbeck and Reyne, an even in which the popular song "Rains of Castamere" was written for.
Tywin was eventually made hand for King Aerys Targaryen and held the year for twenty years of peace and plenty. However several disagreements and jealous from the King eventually drove a rift between the two. During Robert's Rebellion Tywin took no side, not until the very end when he entered King's Landing and sacked the city. He eventually arranged a royal marriage for his daughter, Cersei Lannister with King Robert Baratheon.
Tywin would lead House Lannister though the War of the Five Kings against House Stark and Renly Baratheon. After some early setbacks, Tywin wass forced to engage Renly Baratheon in the Crownlands, as the Siege of King's Landing quickly commenced after Renly's victory over Stannis Baratheon.
Tywin initially had success against the rebels, delivering a crushing blow to Renly Baratheon and nearly capturing the King in the Battle of the God's Eye. Shortly after Tywin would press on in an attempt to lift the siege of King's Landing, now that the odds had been slightly evened. He was unsuccessful, his death during the battle putting his army to rout despite delivering tremendous casualties to the enemy.