Carthage was originally a Phoenician colony. From around 1100 - 600 BC, the Phoenicians were perhaps the greatest sea-faring people in the world. Their original territory was located in modern-day Lebanon, but they very early on began establishing colonies throughout the Mediterranean. The most notable of these colonies was Gades in Spain (now called Cadiz) and Carthage in northern Africa. The Phoenicians themselves were later conquered by the Assyrians and then subsequent empires, but their colonies lived on independent.
Carthage itself was supposedly founded by the mythical Elissa (usually called Dido by the Romans). Because of its favorable position in the Mediterranean, Carthage soon became of a hub of trade which stretched from Spain and Britain in the west to the east and the now developing Silk Road. Soon, Carthage had developed a sizeable empire, although she was always threatened by Libyan nomads in the deserts to the south.
Carthage was jointly ruled by a king and a council of elders. The kings however gradually lost power to the council and to generals, who typically employed large numbers of mercenaries.
For most of its history, Carthage was locked in a struggle for power with the various Greeks states, which controlled the eastern Mediterranean. They had at one time allied with the Etruscans, the Gauls, the Persians, and finally Rome. When Rome and Carthage at last successfully repelled the Greeks, the two victors then came into conflict, which proved Carthage's undoing.
Through three Punic Wars (Poenus is the Roman word for Phoenician) Carthage fought well but Rome steadily increased her power, until Carthage was utterly destroyed in the Third Punic War.