The unnamed son of Cersei Lannister and Robert Baratheon is an unseen character. He is deceased when the events of the series begin and is not expected to appear. He is first mentioned by Cersei Lannister in "The Kingsroad".
Cersei's black-haired son was her firstborn child, and the only one of her children whose father was King Robert Baratheon. Very little is known about him other than 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 the point of his demise are unknown. Had the boy lived, he would have been first in line to ascend the throne when Robert died.
When Catelyn Stark watches over Bran, who has not woken up after having fallen from the tower, Cersei comes to visit. She tells Catelyn that years ago she lost her first boy, a little black-haired beauty. She and Robert were nearly driven mad by the child's death.
Robert and Cersei talk about the imminent Dothraki's invasion and their unhappy marriage. Cersei says that she had felt something for Robert once, even after they lost their first boy, for quite a while.
In the booksEdit
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 loathed 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 that could end the pregnancy. Robert was unaware of both the pregnancy and the abortion.
The TV series invented that Cersei actually had a son with Robert who died in the cradle (apparently to give her a sympathetic scene with Catelyn in the second episode); some viewers wondered if she made the story up to manipulate Catelyn, but later in Season 1 she speaks about the dead baby with Robert himself, which confirms she wasn't lying.
The TV series never mentioned this extra son after Season 1. In subsequent seasons Cersei would refer to Joffrey as her "eldest son", possibly raiding a contradiction. Moreover, in the Season 5 premiere, "The Wars to Come", Maggy's prophecy to young Cersei stated she would have only three children (as it was phrased in the novels). In the novels, however, infant mortality is so high even among the nobility that characters frequently only count the number of their children who survived the cradle. For example, Doran Martell actually had two other brothers before Oberyn was born, but both of them died in the cradle, so they often aren't included in the number of siblings he had.
Writer Bryan Cogman was directly asked about this with regard to Maggy's prophecy, and he indeed confirmed that Cersei's black-haired son with Robert simply isn't included in her "official" count of children because he died in the cradle: "Maggy’s just speaking of the three official kids who lived and were known, etc. The black haired baby was kept quiet."
It is still possible that the writers frequently forgot about this change they introduced in Season 1 - it raises some issues with Joffrey's age - but otherwise it is largely irrelevant to the dialogue about Joffrey being Cersei's "eldest" son, or that she would have "three" children, as it there is an easy explanation to reconcile it. As for Joffrey's age, he is stated to be 17 years old in Season 2, which is also stated to be "18 years" after Robert's Rebellion (meaning he was 16 in Season 1, which was changed to be 17 years after the rebellion instead of 15 as in the books). While stretching the limits of internal consistency, it is still possible that Cersei immediately got pregnant with the black-haired son after marrying Robert at the end of the war, then immediately became pregnant with Joffrey, so two nine-month pregnancies occurred within 18 months/slightly less than two years - which would result in Joffrey still being born in the calendar year after the war ended. It is also possible that the black-haired son died due to being born prematurely, perhaps as early as 32 weeks, giving a few months between the pregnancies. It isn't clear if the writers took into account the previously mentioned black-haired son when they increased Joffrey's age in the TV continuity (in the books, he was born 3 years after the war ended).