Jack the Monkey is Barbossa's devious pet capuchin monkey, and he was named after Jack Sparrow to mock him after being stranded on an island. In Curse of the Black Pearl, Jack is cursed along the with the rest of the crew, and turns into an undead monkey when in moonlight.
He turned back to normal and was free of the curse with everyone else but he then takes another coin and gets the curse back.
"Where is that monkey? I want to shoot something!" -- Jack Sparrow
In Dead Man's Chest, the monkey stays on the Black Pearl and terroirzes the crew. Sparrow enjoys shooting at it to let off steam. In, On Stranger Tides, when the pearl is inside of a bottle from Blackbeard's ship, Queen Anne's Revenge, Jack is miniature and still resides on the Pearl. Sparrow claims he is even more annoying that way.
"Why thank you Jack"
"Oh not you, we named the monkey Jack. [glares at Jack] [Jack squints at Barbossa"