US citizens enrolled in a university degree in Germany do not require a visa but can directly register for a residence permit. One should do so within 2 weeks of arriving by 1) going to the local residents registration office - Bürgeramt - and 2) the aliens registration office - Ausländeramt - where you receive a resident permit (Aufenthaltsgenehmigung). You will ask to get the "residence permit for study purposes" which is valid for 2 years and costs €110.
The documents required to register at the local residents registration office are:
There are offices in any major city - several in the large cities - e.g. in Berlin they can be found here. All details of visa/residence permits for students are also described here. Please contact us with questions.