One of the best characters on Game of Thrones, I always loved Tyrion. When the story starts, he's the Queen's dwarf little brother. When he was born, his mother died in childbirth. Combine that with his "freakish" short height, and he is only loved by his brother Jaime. While not much of a fighter (but stronger and deadlier than you'd expect) he's used his mind to get out of countless pickles.
After being captured, he ends up fighting for his house. Until his father, Tywin Lannister makes him acting-Hand of the King while the family's bannermen are fighting against Robb Stark. One of the few times Tywin has showed any respect towards his hated son.
He does really well as Hand of the King. Due to his leadership, King's Landing is safe from Stannis's armies. Though he is injured and thanked by nobody once the battle is over. His father gives him the "reward" of marriage to Sansa Stark which he doesn't even want. He really wants to marry Shae, whom pretends to be handmaiden.
He has a very bad relationship with not only his father and sister, but his double-nephew King Joffrey. Once the inbred bastard becomes king and Tyrion loses his status as Hand of the King, Joffrey often torments him. When Joffrey is poisoned at his own wedding, everyone is sure Tyrion was behind it all.
I thought his best moments was during his trial. When he demands a trial by combat after Shae betrays him, is one of the best speeches on the show.
And when his champion Oberyn Martell dies, Tyrion is set free by his brother Jaime. Instead of sneaking away, he murders Shae (who he didn't expect to find in his father's room) and shoots his own father with a crossbow. There was a fan theory that Tyrion was the secret son of the Mad King, but it doesn't appear to be so, now that the show is over.
He escapes to Essos and has an adventure with Jorah Mormont. He eventually gets to meet Daenerys Targaryen, which he claims to be "her gift" from Jorah Mormont. He quickly is accepted into her circle of Targaryen loyalists.
And he is named her Hand of the Queen. Though he is less successful at his job here than he was at King's Landing. Because of his advice, Cersei gets the upper-hand in many battles, and the whole world suffers from it.
He resigns as Hand of the Queen and sentenced to death by Daenerys for freeing his brother before the final battle.
But luckily for him, Jon Snow assassinates the Dragon Queen, and he is able to talk his way out of death again. He even nominates the new king, Bran Stark during his would-be execution.
And Bran Stark names him as Hand of the King to "fix his mistakes" as Hand of the Queen. I'm happy with the way Tyrion's story ended.