Baratheon (son of Robert)

"Such a little thing. A bird without feathers. They came to take his body away and Robert held me. I screamed and I battled, but he held me. That little bundle. They took him away and I never saw him again."

- Cersei Lannister

Cersei's black-haired son is an unseen character in Game of Thrones. He is deceased when the events of the series begin and is not expected to appear in the series. He is first mentioned by Cersei in "The Kingsroad".

Background
Cersei's black-haired son is the only of her children whose father is Robert. Very little is known about him: that he had black hair; resembled Robert a lot; and died as a result of a fever. His name, year of birth and age at his demise are unknown.

Season 1
When Catelyn Stark watches over Bran, Cersei comes to visit. She tells Catelyn that years ago she has lost her first boy, a little black-haired beauty. She and Robert were nearly driven crazy by the child's death.

Robert and Cersei talk about the imminent Dothraki's invasion and their unhappy marriage. Cersei says that she has felt something for Robert once, ever after they lost their first boy, for quite a while.

In the books
In the A Song of Ice and Fire novels, Cersei never gave birth to any of Robert's children. When Eddard Stark confronts her about the identity of the father of her children, she admits that Jaime is their father, and that she loathes Robert since their first night, for coming to their bed drunk and calling her by Lyanna's name. She also reveals that Robert got her pregnant once, and her brother found a woman for cleansing her. Robert was unaware of both the pregnancy and the abortion.

In the book "The Lineages and Histories of the Great Houses of the Seven Kingdoms" there is a record of the marriage of Tya Lannister and Gowen Baratheon, about ninety years ago. Their only issue, an unnamed boy, who is described in Malleon’s tome as a large and lusty lad born with a full head of black hair, has died in infancy. Cersei's black-haired son in the show may be loosely based on that child.