EU, EEA and Switzerland
If you are going to stay in Austria for more than three months you have to register with the competent residence authority (Magistrat, Bezirkshauptmannschaft) within four months of your stay and you will receive a confirmation of registration.
Further information can be found here.
Nationals of third countries
After entering Austria with a "visa D — Erwerb" you can apply for the residence permit at the competent residence authority (Magistrat, Bezirkshauptmannschaft). Information can be found here.
Renewal of your residence Permit
You have to apply for an extension before the original permit expires. Information can be found here.