Although Tywin raised Tyrion as his own (albeit begrudgingly), a number of fans believe that King Aerys II — a Targaryen — raped Tywin's wife, Joanna, on their wedding night and maintained a secret relationship with her for years. This illicit union supposedly resulted in baby Tyrion, who Joanna died giving birth to. It's no secret that Tywin hated his youngest son for killing the woman he loved so deeply, but maybe he hated Tyrion even more because he knew that Aerys was his father. This theory would also support Tywin's decision to join Robert Baratheon's fierce rebellion against the Targaryens that nearly wiped out the entire Targaryen line.
It's worth noting that Tywin's grudge against the Mad King can also be explained away by saying that Aerys was extremely paranoid and out of his damn mind (hence his nickname) and had a habit of embarrassing Tywin publicly. Tywin was ruthlessly loyal to his own family, so if he thought for a second that delivering King's Landing to Robert would serve them better than defending it with Aerys, it wouldn't have been out of his character to do so.
