It's one word doing the work of three different ideas.

Three meanings, one word

“Vatican” is used for the physical territory, the Church's governing authority based there, and — loosely — the pope's administration in general.

Vatican City: a country you can walk across in an hour

As a sovereign state, Vatican City is smaller than most public parks, with its own postal system, license plates, and a resident population of under a thousand.

Why it's separate from Italy

It became fully independent in 1929, specifically so the Church's leadership could operate without being subject to any national government, including Italy's.