Austria is a Federal Parliamentary Republic. That means:
Federal: Republicans and conservatives don't like it
Parliamentary: Run by a parliament, not unlike the British Parliament
Republic: Indirect democracy.
The official language of Austria is German but the German spoken in Austria is a dialect and has different vocabulary than "High German", to the point where a native of Northern Germany would have a bit of trouble in Vienna. The correct way of describing the official language is "Austrian German", like "American English" (like how "fag" has different meanings in Britain and USA).
Relations with Germany
Austrians speak German but are culturally different from Germans and Austrians don't want to be part of Germany. Austrians were forced to be part of Germany for seven years under Hitler but Austria became an independent country again after the Second World War and the Austrians have never wanted to lose that independence since then. Austria is now part of the European Union and Austrians are willing to pool sovreignty with others in Europe but they won't pool soverignty with Germany. Austrians are culturally more like Southern Germans like Bavarians but the Austrians feel that they are different from Northern Germans in places like Hanover and former Prussia.