Africa was a province of the Roman Empire. It roughly comprised the territory of present-day northern Tunisia, as well as the Mediterranean coast of modern-day western Libya along the Syrtis Minor. The Roman administrative province is shown, although in Carthagian times the province was larger.
It was the site of the ancient city of Carthage as well as other large cities in that era, such as Hippo Regius (today's Annaba, Algeria). Most of the province fell into Roman hands at the end of the Second Punic War, with the entire province conquered in the Third Punic War.
Most likely the citizens in 'civilized' Rome began calling the entire northern African coast Africa, eventually spreading the name to the continent.
The Arabs later named roughly the same region Ifriqiya, a rendering of "Africa."