There is not much in-game information regarding One-Eyed Jack, only that he is a bandit leader in Rust Commons West. His bandits are unusually well armed, reportedly being supplied by a source other than Marcus Kincaid. It is later revealed that One-Eyed Jack and his crew have been discretely receiving arms shipments from Torgue.
The Vault hunters encounter One-eyed Jack while completing side missions from Marcus. After completing Firepower: Market Correction, and having destroyed a number of bandit ammunition caches in the process, Helena Pierce fears One-Eyed Jack will attack New Haven in revenge. She responds by sending the Vault hunters to launch a pre-emptive strike on One-eyed Jack in order to prevent her fears from coming true.
"You woke the wrong dog!" - This quote is from the DLC Mad Moxxi's Underdome Riot and is probably a glitch as this is originally Nine Toes' quote.
He looks similar to a Psycho, although he uses his gun, Madjack, rather than melee weapons.
A 'one-eyed jack' is a common playing card term used to refer to two of the jack cards which show the face in profile, thus presenting only one eye. In poker, these are frequently set as wild cards.
Unlike other bosses, it's possible to end the fight with Jack quickly by running him over with a vehicle. On the other hand, Jack can be a dangerous opponent to fight on foot because of his high-damage revolver.
The item "Jack's other eye" is represented in game by a Skag pearl.