Daemon I Blackfyre

Daemon Blackfyre is an unseen character in Game of Thrones. He is not expected to appear in the series, being long dead by the time it begins.

Background
Daemon Blackfyre was the legitimized bastard son of King Aegon IV Targaryen and his cousin, Daena Targaryen. After his legitimization, he founded his own new noble House as a cadet branch of House Targaryen: House Blackfyre, named after the ancestral Valyrian steel sword of House Targaryen which his father gave him. He led a rebellion against his half-brother King Daeron II Targaryen, the legitimate Targaryen heir, but was defeated.

With Serena Blackfyre he fathered the twins Aegon and Aemon Blackfyre, and Daemon II Blackfyre.

Season 1
Daemon, along with his bastard siblings is mentioned in the House Targaryen's entry of the book The Lineages and Histories of the Great Houses of the Seven Kingdoms. The tome mentions that he rose in rebellion against his half-brother but perished from wounds sustained at the Battle of Redgrass Field.

In the books
In the A Song of Ice and Fire novels, Daemon is mentioned several times. More of his backstory is revealed in The Tales of Dunk and Egg.

After the Dance of the Dragons, the civil war between Rhaenyra and Aegon II Targaryen, House Targaryen's royal succession laws were revised to put female heirs behind all possible male ones - specifically that if a all of a lord's sons die, the throne will pass to his younger brother before it passes to his own daughters. Under the normal succession laws of the Andals, a daughter inherits before an uncle. Both Rhaenyra and Aegon II died in the Dance, but were succeeded by Rhaenyra's son Aegon III. His two sons Daeron and Baelor became king after him but died without issue. Aegon III did have three daughters, but at this point the new succession laws came into play, so Baelor was succeeded by Aegon III's younger brother Viserys II, who in turn was succeeded by his own son Aegon IV.

Had the new succession laws not gone into effect, Aegon III's daughters would have inherited ahead of Aegon IV, instead of being skipped over for their uncle Viserys II. As it happened, the youngest of these three daughters, Daena, had an affair with Aegon IV, her own first cousin, which produced their bastard son Daemon Blackfyre.

The result is that Daemon and the rest of House Blackfyre are actually descended from older brother Aegon III, while all subsequent members of the main line of House Targaryen (i.e. Daenerys) are descended from younger brother Viserys II.

Daemon was a bastard, and though legitimized, he was simply younger than Daeron II. Therefore, part of his claim to the throne rested on the flimsy accusation that Daeron II was himself a bastard, fathered by Aegon IV's younger brother Aemon the Dragon-Knight. However, others pointed out that Daemon's mother Daena would have been ahead of Daeron II's grandfather Viserys II, had the succession laws not been arbitrarily changed.