If your an American citizen you have it pretty easy travelling around the world, especially to Europe. Your passport acts as a tourist visa and is valid as such for up to a three moth stay. No other paperwork required. However, after three months in Germany you need to obtain a residence permit (Aufenhaltserlaubnis) to legally stay in the country. This permit is obtainable at the Ausl�nderamt (office for foriegners) of the city you are living in. The building in wich the Ausl�nderamt is located houses many other goverment officies, including the drivers license bureau and Arbeitsamt (employment office). The names of these buildings are different from city to city so check the websites for the city you want to live in and they should have the correct information on the name of the building and its location.
When you want to apply for a residence permit in Germany you need a couple of things. Of course your passport (never leave home without it if you plan on visiting a goverment office), proof of health insurance (AOK is a common one for students), proof that your living in the city (a confirmations certificat from the Einwohnermeleamt, will be discussed later), proof of sufficient funds, (i.e. bank statement or a letter of financial support from guardians) and proof that you have either 1. A permanent job, or 2. a place in a University to study, participating in a language school is also valid.
After moving into your first apartment in Germany you need to register at the Einwohnermeldeamt, which may or may not be located in the same building as the foriegner office. To register to you need your passport and your rent contract (mietvertrag) from your landlord. This registration is necesarry if you want to renew your visa. Also, if you move you should update your adress with the Einwohnermeldeamt so your location is always current. This not only helps the Goverment keep track of all its citizes and foriegners living there, but also is used to charge fees for the use of its "public TV and radio". The GEZ (Geb�hreneinzugszentrale) does a good job finding you by acessing the records at the registration office. IF you have a TV or radio in the house you need to pay a fee for using them, which ammounts to around 45 euros for three months. Also, IF you tell the GEZ that you have a TV or radio, you MUST go to the registration office (Einwohnermeldeamt) to unregister your apartment, otherwise the GEZ will still charge you for the TV and radio because your registered there, even if your not living at that address. Additionally, you have to unregister anyway even if you dont pay anything to the GEZ since when you apply for your residency permit have to have proof that your actually have an apartment in the city your applying for residence.
This is the basics of the German beuracracy for Americans and many foreigners for obtaining a visa. If you have any unanswere questions turn to the American embassy website, they have alot more info than is posted here. I just wanted to give a consice list of what was entailed here.