The Holy See
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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.
Holy See Coat of ArmsMeaning of the Holy See Coat of Arms
Since the XIV Century, the two crossed keys have been the official insignia of the Holy See. The gold one, on the right, alludes to the power in the kingdom of the heavens, the silver one, on the left, indicates the spiritual authority of the papacy on earth. The mechanisms are turned up towards the heaven and the grips turned down, in other words into the hands of the Vicar of Christ. The cord with the bows that unites the grips alludes to the bond between the two powers.
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.
Coat fo Arms Information Source:
http://www.vatican.va/news_services/press/documentazione/documents/sp_ss_scv/insigne/sp_ss_scv_stemma-bandiera-sigillo_en.html#Stemma della Santa Sede
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.