Carthage was an ancient city. It was a lot like Rome, except it was in Africa. Because the Romans burnt Carthage to the ground anyway, it can be reasonably assumed to have never existed at all, just like the rest of Africa.
Hannibal was from Carthage. He probably started the war with Rome because he ate too many Romans and Julius Caesar got angry, or something like that. The Sicilian Mafia may or may not have been involved.