If you stay in Germany for more than 90 days, you need to apply for a residence permit with the local Foreigners Office. You will need an appointment in order to do so and to get to know what documents you have to take with you. It is advisable to book the meeting soon after arrival as the procedures of getting the residence permit might take some time.
The application for a residence permit will cost ca 100 Euro. Stipend holders do not need to pay this fee.
Ask the WO 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 assistence at the Office.
EU citizens will receive a confirmation about their right to stay (so-called “EU-Freizügigkeitsbescheinigung”), generally automatically some weeks after registration. This confirmation is a work permit at the same time and usually has unlimited validity. Therefore, it is neither necessary nor possible for EU citizens to apply for a residence permit as mentioned above.