Captain Hector Barbossa is the overall deuteragonist of the first five Pirstes of the Caribbean franchise. He was the (former) main antagonist of the first Pirates of the Caribbean film, The Curse of the Black Pearl. cameo character in Dead Man's Chest and a character in the last three Pirates film.
Prior to the series, Barbossa was once an honest sailor, who fled from his father (who berated him) at a young age. He was the First Mate and a friend/rival of Jack Sparrow. Barbossa planned a mutiny with the other members of their crew when they saw that Sparrow was unfit to be their captain and marooned him on an island with only a pistol and a shot.
Soon, the pirates were cursed because they would not return the treasures that belonged to Hernan Cortez. He was also the Pirate Lord of the Caspian Sea, having been given a wooden eye that he entrusted to Righetti.
The favors Barbossa has done for Jack is killing Pig Kelly and saving him from Davy Jones' locker.
Johnny Depp and Geoffrey Rush both teased that "Hector" was Barbossa's first name, as his name was not established in the film itself but on the DVD commentary of The Curse of the Black Pearl. The directors also thought that Barbossa's first name was Fan-made. Since Depp, Rush and the fans called Barbossa this name, Hector was officially made the treacherous captain's name.
It is spoken first in At World's End by Jack Sparrow, further confirming Hector is his first name.
Barbossa naming his monkey, Jack, after his former captain, Sparrow, is similar to Long John Silver from Treasure Island naming his parrot, Captain Flint, after his former Captain