You Need To Specifically Register For German University Exams
When you register for a college or university course in the U.S., do you expect to have to register for the exam separately from registering for the course you're taking? Of course not! The registration for the course and the exam are one and the same. Depending on what and where you are studying in Germany, this is likely not to be the case however.
For the typical German university course, you don't necessarily have to register to attend the course, unless there is a practical component, additional material online that you need to have access to, or you are going to participate in a seminar and need to give a presentation. It's rather common for students who have to pick between taking a number of lectures for a given semester to start off attending all of the lectures, and drop the ones that they don't need or don't need to take just yet in favor of attending others.
At the mid-late section of the semester, a notice for exam registration will typically be emailed out (check the email address provided to you by the university!), along with information on how to register on an online portal. Your login information for the portal is usually given out a the start of your studies (which will be in your entrance paperwork for those only studying abroad for a semester). Once the registration timeline closes, you are on the hook for taking the exam, and missing the exam will result in you receiving a 5.0 (fail) on the exam.
For those taking primarily language courses, or certain specific degree programs which have a completely fixed course track, registration for the exam at the end of the semester is likely compulsory, and you may already be registered from the beginning of the semester. If you are attending more courses than you need credits for based on your obligation to your home institution, double check what your given course's requirements are, so you don't find yourself ending up with a grade for a course you didn't mean to register an exam for.
No matter what the specifics of your exam registrations, you will need to get a faculty specific transcript issued to you. The traditional method for issuing transcripts is to provide students with a certificate (Zertifikat) for each exam taken, which will list the course, grade, and number of ECTS points (credits) issued for the particular lecture, seminar, or combined module you've received. This means for example, that if you take a math course, language course, and history course, you will need certificates from either each faculty or professor directly.
If you need one specifically from a professor, go to their faculty's student office's open hours, and ask for a certificate to be issued to you. In the event that you will be leaving from Germany prior to the grades being made available, as can happen in some cases, you may have to provide them a self-addressed envelope (with enough postage for the U.S.!) to make sure it reaches you. In general however, this is unnecessary, as the student offices are fairly good at helping to expedite the process for international students that are leaving the university at the end of the semester. That being said though, make sure that you don't wait until the last possible moment, as at the least, the professor will need to sign off on the certificate, and their hours in the office outside of lecture times can be more sporadic than you might think.