The Purple Wedding
Joffrey Baratheon chokes to death at his own wedding feast, poisoned in front of several hundred witnesses, almost all of whom are pleased. Tyrion is blamed because he is the most convenient person to blame, and the trial that follows finally breaks whatever was left between him and his father.
The fuller tale
Joffrey Baratheon married Margaery Tyrell in the Great Sept and held a wedding feast of seventy-seven courses in front of a thousand people, most of whom disliked him, and used the afternoon to humiliate his uncle in public one last time.
He choked on the dais, in front of everyone, clawing at his own throat with his mother screaming and his new wife's family watching, and died pointing. What he was pointing at is a matter of opinion. What he had been drinking was poured by his uncle, who had every reason and had nothing to do with it.
The poison had come off the hairnet of an old woman who was there for exactly this purpose, dropped into the cup by a fool who was not a fool, arranged by a man who then removed the bride's sister from the city in the confusion. Nearly everyone at that wedding wanted the boy dead and only three of them had done anything about it.
The trial
Tyrion was tried because he was convenient, and because his sister had wanted him convicted of something for thirty years. The evidence was assembled out of things he had actually said ÔÇö every bitter joke he had ever made in earshot of a servant, read back to him without the room around it.
The witness who broke him was the woman he had tried to protect. He asked for trial by combat and the man who fought for him won and then insisted on talking, and lost. Everything Tyrion does after that night he does as a man with no family left to disappoint.
Elsewhere in the wiki
 See the world as it stood in 300 AC What became of it  if an ending is written, it is here. Spoilers.
The chronicle records no ending here ÔÇö or none the maesters have set down.