- "Argilac and his men charged and the Baratheon line broke, revealing Queen Rhaenys and Meraxes. Argilac's vanguard burst into flame and his men panicked. The victory charge fell into chaos and Argilac himself was thrown from his horse, but he didn't yield. When Orys Baratheon arrived, he found the old king holding off half-a-dozen men, another half-dozen dead at his feet. Orys dismounted to meet the king on equal footing and offered Argilac one final chance to yield. Argilac cursed him instead. As the storm raged around them, the grizzled old warrior and the bearded upstart fought for life and kingdom. In the end, Argilac got his wish; he died sword in hand."
Even though the massive walls of Storm's End provided protection against Baratheon's siege, Argilac decided to meet the invading armies in open battle after hearing about how Aegon Targaryen's dragon Balerion destroyed Harrenhal. Orys Baratheon and the Targaryen army were victorious and Argilac was killed.
As a reward for the victory, Aegon legitimized Orys Baratheon and named him Lord Paramount of the Stormlands. To cement his rule, Baratheon took Argilac's daughter, Argella, as his wife. He also took both the sigil (a black stag on yellow), and words ("Ours is the Fury") of House Durrandon as his own: thus, House Baratheon was founded.
In the books
Argilac was said to be arrogant for engaging the Targaryens in open battle rather than withstanding a siege in the infamously impregnable castle of Storm's End. However, Argilac was motivated in this by having heard of the death of Harren Hoare in similar circumstances at Harrenhal, who prepared for a siege in his immense castle only for the Targaryen dragons to fly right over the walls. If Argilac resisted a siege by Orys it would only be a matter of time until the Targaryens directed their dragons to the castle, so wishing to avoid this fate Argilac met Orys in pitched battle. It is possible that Argilac knew he couldn't win, but simply wanted the dignity of dying while fighting, not burned out of his castle like a rat.
Orys's army suffered a defeat on the march to Storm's End but was able to recover with minimal losses and continue. Orys was accompanied by Aegon's sister-wife Rhaenys and her dragon Meraxes, though because Rhaenys was not as militaristic as her sister Visenya or as skilled as a general, Orys remained in command of their army.
Despite Aegon's success at Harrenhal, the battle against Argilac was grueling: he had the sense to sally forth to attack during one of the frequent lightning storms which blew in from Shipbreaker Bay, during which it was too dangerous for Meraxes to fly. Orys was outnumbered two-to-one, and at one point was outflanked, but then Rhaenys plugged the gap riding Meraxes on the ground - even denied the advantage of flight, the fire-breathing dragon was still formidable. The battle culminated when Orys faced Argilac in single combat and slew him on the battlefield.
Afterwards Argilac's daughter Argella barred the gates of the castle, proclaimed herself Storm Queen, and vowed to fight on, but she was betrayed by her followers and handed over to Orys naked and in chains. Orys treated her with courtesy, freeing and feeding her, and accepted her surrender. Orys later married Argella to cement his control over the Stormlands, and in the process founded House Baratheon.