The Pope is the Catholic bishop and patriarch of Rome, and the visible head of the Catholic Church throughout the world. In addition to this spiritual role, the Pope is also head of the independent, sovereign State of Vatican City, a city-state entirely surrounded by the city of Rome. Prior to 1870, the Pope's temporal authority extended over a large area of central Italy, a territory formally known as the "Patrimony of St Peter" under the terms of the Donation of Constantine, but known more familiarly as the Papal States. The office of the Pope is informally called the Papacy and formally called the Pontificate; his ecclesiastical jurisdiction is called the Holy See (Sancta Sedes). Roman Catholics worldwide consider each pope to be Jesus' representative on Earth. The current pope is Benedict XVI.
The name "pope" is the English form of the Latin "papa" which means "father" and was a form of address for bishops in the early church (contra Matthew 23:9). By about 500 AD the term was being applied mainly to the Bishop of Rome until, in 1073, Gregory forbade Catholics to call anyone "papa" except the bishop of Rome. At this stage one can speak of a single "pope" as we know today albeit in a much different form as many powers and privileges were bestowed as time progressed.
The role of Peter
Catholics are taught that Peter was the first Pope followed by Linus whereas the historical evidence provides no basis for Peter ever having been a bishop, nor was he in Rome for any length of time. Indeed Irenaeus names Linus as the first bishop of Rome. The link to Peter appears to be an attempt to benefit from the authority bestowed on Peter in Matthew 16:18 (the Rock on which Jesus will build his church). Whilst Peter's apostolic authority is evident in the New Testament, nowhere does he assume a special leadership role above other apostles nor is this otherwise indicated.
The broken line of popes
The history of the papacy is indeed a regrettable account of intrigue, corruption, political maneuvering and even violence. For many years the pope was more powerful than the emperor. Papacies were bought, murdered for and even run in parallel where disputes could not be resolved.