Persons who are not citizens of an EEA country or Switzerland (i.e. third-country nationals) and plan to stay in Austria for a period of more than 6 months require a residence title.
For stays of up to 6 months within one year, a visa must be applied for (see Visa).
Although EEA citizens wishing to stay for longer than 3 months in Austria do not require a residence title, they must apply for a registration certificate with the competent authority within 4 months after having entered Austria. More info here.
Under Austrian law there are different types of residence titles:
A person intending to stay in Austria for a temporary period of time only requires a residence permit.
A person intending to permanently stay or settle in Austria requires a residence title which qualifies as settlement under the terms of sec 45 of the Settlement and Residence Act. This includes settlement permits, the Red-White-Red Card, the Red-White-Red Card plus, the Blue Card EU and the Permanent Residence – EC residence title.