Residence Permit

If you stay in Germany for more than 90 days, you need to apply for a residence permit in the local Foreigners Office (“Ausländerbehörde”). It is recommended to make an appointment in order to do so and to consult the International Office for the necessary documents in advance. It is advisable to book the meeting soon after arrival as the procedures of getting the residence permit might take some time.
All facts about the Electronic Residence Permit

Ask the International Office if you have questions or need any kind of help, such as booking the appointment, getting a confirmation letter about your stay and payment or assistance at the Office.

In short:

You do not need a residence permit, if you are a citizen of the European Union.

All other foreign guests have to apply for a residence permit:

before their visa expires;
if they came to Germany without a visa.

The following items are requested:

application form
two biometric passport photos
your valid passport
scholarship/MPI- contract
accommodation registry
copy of your "Anmeldung" (see registration)
copy of your health insurance certificate (see health insurance)
if you are enrolled at a German University: proof of enrollment
proof of having successfully finished university etc. (copies).

If your spouse and/or children are with you:

proof of health insurance for both
German translation of certificate of marriage (for Chinese nationals: legalized marriage certificate)
children need their birth certificate.