Each country can make their own rules for what is required for other citizens to live inside of Austria. In many countries to live for an extended period of time, you will need to acquire a visa. You can do that by calling the government department associated with immigration and visas. To come into the United States whether it is just to live, work or go to school, you will need a visa.
However, if you are a U.S. citizen, then you do not need a visa to go to Austria. There are limitations though. First, you may go to Austria without a visa as long as you are only going to stay ninety days within a one hundred eighty day period.