Aerion Targaryen is a character mentioned in the third season episode "And Now His Watch is Ended". He is long dead by the time of the series and only appears in Histories & Lore. He was a Prince of the Targaryen dynasty.
Despite the fact that both his father and grandfather broke with the Targaryen custom of marrying brother to sister "to keep the bloodline pure", Aerion still succumbed to the madness that plagued many members of his family due to multiple generations of compound inbreeding. He ultimately killed himself by drinking wildfire, believing it would transmute him into a dragon.
Touring the Great Sept of Baelor in King's Landing, King Joffrey Baratheon points out to Margaery Tyrell the urn containing what little remains of Aerion Brightflame. He briefly explains who he was and the manner of his death.
In the books
In the A Song of Ice and Fire novels, Aerion Targaryen is remembered as an example of the madness that can befall the members of House Targaryen, believing himself a dragon in human form, and had a rather cruel streak that would lead him to be nicknamed Aerion the Monstrous. In his youth, Aerion was slim and of average height with curly silver-hair, a statuesque face of sharp cheekbones, straight nose, pale skin, and violet eyes.
Aerion's brothers were not fond of him, particularly Aegon. At some point of Aegon's childhood, Aerion threw his cat down a well and stole into his chambers with a knife and threatened to castrate Aegon to get a sister he could marry. After the events of the Ashford Tourney, which led to the death of Maekar's brother Baelor, Aerion was sent to Lys, where he may have fathered some bastard children. During his exile he also served with the Second Sons, a mercenary company.
In all fairness, Aerion did not simply have a fit of madness one day in which he decided that drinking wildfire would turn him into a dragon: he was actually very drunk at the time. He was always quite arrogant and unstable, but under normal circumstances was not outright delusional.
Aerion married his first cousin, Calla Targaryen - the only surviving child of his uncle Rhaegel Targaryen, who was widely believed to have been insane (not sadistic like Aerion, but is said that he was seen dancing naked through the Red Keep). Aerion had one child by Calla, a son, that in a display of pique and arrogance he named "Maegor" - after his ancestor, the infamous tyrant King Maegor the Cruel. The first Maegor's memory was so despised that the Targaryens had avoided naming any of their sons "Maegor" again for the previous two hundred years (in real life terms it is comparable to how the English monarchy never named a son "John" again after King John, brother of Richard the Lionheart, or like naming a prince "Nero" after Emperor Nero).
Aerion died drinking wildfire in 232 AC, and his father Maekar died only a few months later that same year. Aerion's son Maegor was still just an infant, and it was feared by many that the son had inherited the father's madness (and possibly also his mother's father's madness). At the Great Council convened to determine the succession, only a few spoke up for Aerion's son and he was quickly passed over as a candidate. It also passed over the only heir of Maekar's eldest son Daeron, a ten year old girl named Vaella who while sweet was mentally disabled. Maekar's third son Aemon had become a maester, leaving his fourth son Aegon V to inherit the throne (though due to Aegon V's populist leanings there were many among the nobility who pressured that Aemon should be released from his vows to inherit ahead of him, but Aemon flatly refused to have any part in this, and joined the Night's Watch).
Aerion Brightflame is one of the main antagonists in the first entry in the Tales of Dunk and Egg series of prequel novellas set 90 years before the War of the Five Kings. He is a major opponent of Ser Duncan the Tall, and challenges him to a trial by combat - or rather, a rare "trial of seven" variant in which two sides of seven men fight each other. Aerion Brightflame would thus appear in any potential TV adaptation of the prequels.