The Roman Empire was split into Western and Eastern portions in the late 3rd century, with the seat of power for the West being Rome and the seat of the East being Constantinople. The two halves were briefly reunified under Constantine I, but after that they were effectively two separate nations, linked only by history and tradition.
Starting in the mid 4th century, the Western empire was subjected to repeated invasions by Germanic peoples, most violently by the Visigoths and Ostrigoths. In 410, Rome was sacked by Alaric I, King of the Visigoths, and the Western empire was dismantled in 476 when a German mercenary, Odoacer, led an overthrow of Western emperor Romulus Augustus. For nearly two centuries after that, the Italian peninsula was a battlefield between Gothic, Byzantine and Italian forces.
Into the power vacuum stepped the Patriarch of Rome. It is around this time that the Pope assumed the title of Pontifex Maximum, a title held originally by the chief priest of Iupiter and latter held by the Emperors to represent their authority as the gods' divinely annointed representative on earth. It is also around this time that the College of Cardinals begins to take shape, when the now Christianized Collegium Pontificum (originally, an organization made up of the highest ranking priests and priestesses of pagan Rome) and the remnants of the Roman Senate merged and took responsibility over both religious practice and civil law. To this extent, the Catholic Church is, indeed, the inheritor of the Western Empire.