Short answer: The Vatican is a country.
Long answer requires understanding that there is a difference between "State" and "Country", but The Vatican qualifies as both anyway.
The Vatican City is a State (the legalistic definition of an independent country), meaning it has full national sovereignty. It makes its own laws, has citizens, some of whom live there permanently, it issues passports, it has an economy and a national bank, it issues currency (the Euro), it has territory (land), it maintains diplomatic relations at State party level with other nations, it has international borders, a national government (in the Vatican's case, a theocratic monarchy), and it is recognised as a State by other States. It maintains post and telephonic services, has (a very limited) public transport system, its own national military service, and so on.
It's important to note that the Vatican city (the national entity) is separate to the Holy See (the religious entity) which maintains its own relationships with other entities.
Also, someone mentioned that The Vatican is the world's smallest country. In fact, that dubious honour goes to Sealand on the basis on having the smallest land area, though not many other States recognise Sealand as sovereign.
And now the extended answer:
The Vatican City also qualifies as a country. Countries do not have to be sovereign. For example, Puerto Rico is a country but not a State because it is administered by the United States, does not regulate its own economy or issue money, enjoys limited domestic autonomy (its capability to make laws is limited by the US federal government), is not recognised by other States as being a State, and so on. In this respect it is similar to other sub-national entities such as Hong Kong (part of China) or England (part of the UK) or the internal states of the United States.