The pope is generally not called the Holy See, he is the bishop of the Holy See. The pope lives in the Papal Apartments located in Vatican City, and has a summer residence in Castel Gandolfo.
- The pope is what we call the successor of St Peter: he is the bishop of a very old diocese (or "see", or "seat") called the Holy See. As its bishop, he has jurisdiction over that diocese. But because the pope has "primacy", he also has a certain kind of jurisdiction over the entire universal Church.
- Vatican City is a sovereign city-state, established in 1929 and located inside the city of Rome. It is one of only three city-states today; Athens and Sparta are well-known ancient examples. Its ruler is the pope.
- The Holy See is the diocese of the pope. It is also called the See of Saint Peter, the Apostolic See, and the Diocese of Rome. Its territory includes both Rome and Vatican City. This is an important diocese because it is the diocese of the pope, assisted by the Roman Curia (the Court of Rome), who together form the main governing body of the Church. The Holy See is a sovereign entity.
The Holy See is a sovereign entity, but is not a nation: it is a diocese that has universal jurisdiction. The territory of any diocese aligns with national/political borders only for practical reasons. If a church building were able to separate its church grounds from its country, it might become a sovereign state, but it would still be a part of its diocese, under the authority of its bishop.
When someone says "the Vatican" they might be talking about either the Holy See or Vatican City. Note the Catholic Encyclopedia article on the Holy See, part of which says:
[...] We can thus readily understand how Holy See came be the technical term for the pope, the central ecclesiastical government, and the actual abode of the same.
It is probably best not to copy this potentially confusing practice (see synecdoche).
For more information see the section on jurisdiction in the Hierarchy article, and the article on the Pope.