Aegon I Targaryen, also known as Aegon the Conqueror was the one (along with his sisters/wives Rhaenys, and Visenya) who united the Seven Kingdoms through the use of of his dragons, Balerion, Meraxes, and Vhagar. This is whom Ser Barristan is referring to.
Aegon has been used numerous times since (at least 12) in Targaryen family. Much like a real royal family, the popular names get reused.
In the show universe there are even two more Aegons we are introduced to, both sons of Rhaegar...
It is possible that Rhaegar was naming his sons Aegon due to prophecy. The quote below is about naming his first born son.
The man had her brother's hair, but he was taller, and his eyes were a dark indigo rather than lilac. "Aegon," he said to a woman nursing a newborn babe in a great wooden bed. "What better name for a king?"
"Will you make a song for him?" the woman asked.
"He has a song," the man replied. "He is the prince that was promised, and his is the song of ice and fire." He looked up when he said it and his eyes met Dany's, and it seemed as if he saw her standing there beyond the door. "There must be one more," he said, though whether he was speaking to her or the woman in the bed she could not say. "The dragon has three heads."
A Clash of Kings - Daenerys IV