Daenerys Targaryen (daughter of Aegon IV)

"For thousands of years the Storm Kings fought the Gardener kings of the Reach and various families of Dorne for control of the Dornish Marches just below the Red Mountains. The fighting did not stop until Dorne married into the Seven Kingdoms a mere hundred years ago."

- Brienne of Tarth

Daenerys Targaryen is an unseen character in Game of Thrones. She is not expected to appear in the series, being long dead by the time it begins.

Daenerys was a Princess of the Targaryen dynasty and the namesake of the later "Queen Daenerys Targaryen, First of Her Name, the Stormborn, Mother of Dragons." She lived about one century before Daenerys Stormborn. Specifically, she was the younger sister of King Daeron II, who was the grandfather of Daenerys Stormborn's own grandfather, King Aegon V. Put another way, she was the paternal aunt of Daenerys's great-grandfather, King Maekar.

Background
Daenerys was the daughter of King Aegon IV Targaryen and Queen Naerys Targaryen, born on the year 182 after Aegon's Landing. She was the sister of King Daeron II Targaryen. She also had a half-brother, Daemon Blackfyre, born of the affair between her father and Queen Daena Targaryen, the sister-wife of Baelor I Targaryen. She married the Prince of Dorne, Maron of House Martell.

Through her brother King Daeron II, Daenerys was the aunt of King Maekar, and thus the great-aunt of both King Aegon V the Unlikely and Aegon V's brother, Maester Aemon of the Night's Watch.

Season 1
Daenerys, along with her family, is mentioned in the House Targaryen entry of the book The Lineages and Histories of the Great Houses of the Seven Kingdoms.

In the books
In the A Song of Ice and Fire novels, Princess Daenerys was born well after her older brother, Daeron II, to the point that Daeron had already married and had a son, Baelor Breakspear. Her betrothal to the Prince of Dorne, Maron Martell, was brokered by her brother Daeron as part of the treaty bringing Dorne under the authority of the Iron Throne. Daenerys supposedly loved her bastard half-brother, Daemon Blackfyre. It was said the main reason Daemon rose in rebellion against Daeron was that Daenerys was denied to him. A year after the rebellion concluded, in 197 AL, the marriage between Daenerys and Maron was carried out and Dorne formally joined to the rest of the Seven Kingdoms.

After Daemon's death, Prince Maron built his new bride the Water Gardens, a place of beauty made of colored marble with pools and trees. Her children played in the pools along with the children of other highborn lords. One day when the sun was hot, she took pity on the children of the servants and guards and allowed them to play in the pools with the highborn children, starting a tradition that is still kept in Dorne to this day.

While her brother's descendant Daenerys Stormborn is the second member of the Targaryen family to be named "Daenerys", Daenerys Stormborn is not called "Daenerys II", because "Daenerys, daughter of Aegon IV" never ruled as a queen. In fact, there has never been a ruling Targaryen queen before. When Daenerys Stormborn proclaims herself the rightful heir of the Targaryen dynasty (after her brother dies), she is officially styled as "Daenerys of House Targaryen, First of Her Name". This has been confirmed to be the same in the TV series continuity, as Xaro refers to her as "Daenerys, First of Her Name" in Season 2's "The Ghost of Harrenhal", and this is also how she is formally introduced throughout Seasons 3 and 4.