How To Register For University Classes
If you are going to be studying abroad in Germany, unless you are part of a program which specifically registers your in advance for specific courses or a plan of study, you will have to register after you arrive in Germany. When you do so, depending on the university and the timing of your arrival, how you go about this will likely be different than you are used to back home.
Most American colleges will have a system with an online registration program, typically that is set up before the previous semester ends. In Germany, for students in a standard course of study, this is also usually the case. If you are coming over for a semester or year, during your first registration period, if you are able to use this system, you should go ahead and do so, as it will make your life that much easier amongst all the registrations that you will have to do at the start of the semester and your time in Germany.
If you are planning to take classes in multiple faculties (i.e. language courses and science courses in the same semester), you'll have to go to the individual faculty office and register in person. This is also the case if you arrive too late to use the online system. The actual process is simple enough, just bring your matriculation documents with you and visit the office during their open hours. If you are tight for time to the start of the semester, go to the office during typical work hours (8-4, 9-5), and ask to speak with the person at the office. They are usually willing to work with you as an international student (but if you show up speaking sufficiently good German, maybe lead with the fact that you're an international student so they don't assume you're a lazy German student who didn't follow proper procedure).
The reason you have to register separately with multiple faculties if you are taking courses from different faculties, is because your registration is under one of the specific faculties. For university students in the U.S., this system may make more sense than those at smaller colleges, who can register for pretty much any course at will. The university won't try to prevent you from taking a course due to your faculty, but it does mean that your user account in the online system won't be set up to show you courses from other faculties.
If you are taking a language course, you'll need to provide proof of your language level, so that you can be sorted into the correct level course. This is likely also going to be the case if you want to take a course that is in German. In my experience, you're going to need to be at least at the B1/B2 level to handle courses that actually take place in German, and it's going to be difficult (maybe limit yourself to seminars). You are going to have to have a certificate of some kind (like one from the Goethe Institute, or from an entrance test at the university you are studying at). Paperwork from your home institution is unlikely to be considered sufficient.