While attending a tournament at Ashford in the Reach, Ser Duncan the Tall, a poor hedge knight, became smitten with a beautiful Dornish woman. However, the arrogant Prince Aerion "Brightflame" Targaryen accosted the young woman, at which Ser Duncan sprang to her defense - as any true knight should - and struck Prince Aerion. To strike a member of the royal blood was treason, and the corresponding sentence would usually be the loss of the offending hand. Ser Duncan, however, demanded a trial by combat, which at the insistence of Prince Aerion then became a rare "trial by seven", involving two teams of seven men each.
Ser Duncan's side was victorious, but during the combat Aerion's uncle Crown Prince Baelor "Breakspear" Targaryen accidentally took a severe head injury by his brother Prince Maekar Targaryen, and the resulting concussion killed him.
In the books
The trial by seven of Duncan the Tall is the central event of the first installment in the Tales of Dunk and Egg prequel novellas set 90 years before the events of the main A Song of Ice and Fire series, and it wasn't just Aerion who claimed he had been wronged by Duncan; his eldest brother Prince Daeron (who so far appears to be omitted from the show) falsely claimed that Duncan kidnapped Aegon, though Daeron later shows regret for doing so.
It was a blow from Maekar's mace which gave his brother Baelor his severe head injury. Baelor had entered the trial wearing his son's lighter armor, which did not fit properly and didn't provide adequate protection. Maekar had not meant to severely harm Baelor, and Baelor seemed fine at first, acknowledging that it was an accident. Soon, however, Baelor started behaving drunkenly, and when his helmet was taken off, part of his skull fell out of it: the blow had caved his head in, and part of his brain was visible. Baelor collapsed and soon died. The guilt of accidentally killing his own brother haunted Maekar for the rest of his days.
The events surrounding the trial also convinced Maekar that Aerion was a psychopath unworthy to be his heir, and that he should focus on the upbringing of his younger son Aegon, then only a small boy better known by his nickname, "Egg". Through a series of events at the tournament and trial Ser Duncan had befriended Egg, and Maekar agreed to let Egg serve as Ser Duncan's squire as he traveled through Westeros, to give the young prince a better understanding of the world. In later years, and several other deaths in the royal family, Egg succeeded his father as King Aegon V Targaryen, and "Dunk" became the Lord Commander of his Kingsguard.