Visas for stays of up to three months are not required for citizens of the EU, the European Economic Area (EEA), much of Eastern Europe, Israel, USA, Canada, the majority of Central and South American nations, Japan, Korea, Malaysia, Singapore, Australia or New Zealand. All other nationalities require a visa; the Ministry of Foreign Affairs website at www.bmaa.gv.at has a list of Austrian embassies where you can apply for one.
If you wish to stay longer you should simply leave the country and re-enter. For those nationalities that require a visa, extensions cannot be organised within Austria; you’ll need to leave and reapply. EU nationals can stay indefinitely but are required by law to register with the local Magistratisches Bezirksamt (magistrate’s office) if the stay exceeds 60 days.
Austria is part of the Schengen Agreement which includes all EU states (minus Britain and Ireland) and Switzerland. In practical terms this means a visa issued by one Schengen country is good for all the other member countries and a passport is not required to move from one to the other (a national identity card is required, though). Things are a little different for the 10 new EU-member states which joined in 2004; a passport is still required to move in and out of these countries, but check with your local embassy for more up-to-date information. Austrians are required to carry personal identification, and you too will need to be able to prove your identity.
Visa and passport requirements are subject to change, so always double-check before travelling.