In archaeological and linguistic usage, it refers to the Greco-Roman era culture and dialect of Carthage and its empire as distinct from their Phoenician originals. Phoenicians settled in Northwest Africa (the Maghreb) from the city of Tyre (in modern Lebanon) and their culture and political organisation separated into a distinct form. There were distinct Punic settlements from the Iberian Peninsula and Gibraltar in the West to Cyprus in the East; Sicily was a battleground between Punic and Roman forces for a long time. In the end, the Romans managed to defeat the Carthaginians and their allies, enabling a Roman settlement of Africa. There is a famous quote associated with this defeat: Delenda est Carthago, meaning simply, "Carthage is to be destroyed".
See Phoenician languages and Carthage for more on Punic.