Even after the curse was lifted in the first movie, Barbosa's monkey Jack remains undead.
In Dead Man's chest, Jack trades the monkey with the witch and shot him and pronounced him as undead.
How is that possible?
At the end of "Curse of the black pearl" after the credits there is a scene where Jack (monkey) takes a coin from the chest getting himself cursed again, similar to what Captain Jack Sparrow did during the fight with Barbossa.
Sparrow took the coin during the fight which he returned later,Jack(monkey) on the other hand didn't return it. He had probably hidden it with his other shiny things or lost it somewhere, he is a monkey after all :/
Have you watched the after credit scene in the first film? That's where the monkey steels another coin.