You will not need a visa to travel to Austria for a week. Any amount of time less than 90 days is allowed with just a passport in hand. If you wish to stay more than 90 days in any 180-day period, however, you’ll need a Schengen visa.
“The Schengen visa is the most common visa for Europe,” explains its website. “It enables its holder to enter, freely travel within, and leave the Schengen zone from any of the Schengen member countries. There are no border controls within the Schengen Zone.”
Note that if you’re planning to stay longer (i.e. more than 90 days) in any one country for work or school, you’ll need a visa from that country.