Matriculating As A Student In Germany
In the U.S., once you start a degree or study program, your continued registration in the program, assuming you are earning passing grades and don't have any other extenuating circumstances, is essentially automatic. Obviously, you still have to pay your tuition and any other fees to continue, but your studentship is not otherwise in question. In Germany, you have to rematriculate every semester, and you're on your own to make sure it happens.
Each semester, as a student in Germany, you have to matriculate (pay your student fee) to continue your student status, and be able to register for the new semester. Paying the semester fee is usually one of the last parts of your initial registration that you will do when you start your program or time abroad in Germany, and if you continue on to further semesters, you only have to pay the student fee to matriculate for the new semester.
When your matriculation is confirmed after the university receives the fee you have transferred to them, if your university sends a new student ID each semester, you should receive it in short order. If your university instead has a plastic card that is stamped or otherwise validated, you should make sure to take care of this before the official start of the semester. In either case, a Matrikulationsbescheinigung (matriculation certificate) should be made available to your (probably through an online portal). This certificate is what you will use to prove your status as a student (i.e. to your insurance company or bank).
In Germany, if you are a PhD student, you are not necessarily obligated to actually matriculate as a student in order to keep your position. You should make sure that you are registered as a student by the university (typically handled in the same way that the registration of university employees into the various computer systems and user accounts), but you don't typically have to actually pay a student fee and matriculate. Doing so though typically allows you to receive a substantially reduced priced semester ticket on the local public transportation, as well as means you will have a proper student ID, which you need in order to get most discounts that are available to university students.
If you are changing degree programs within the same university, meaning that you are starting a new degree, make CERTAIN that you have confirmation of your finalized registration in your new program, particularly if your matriculation number has not changed. In most of the Germany university systems that I have encountered, the matriculation process is automated to recognize your matriculation number, as given in the subject line of the money transfer, meaning that if you have started a new program, you could accidentally end up matriculating for another semester as part of your last program. At the least, this will cause a headache for you and the student office, at worst, this could cause you difficulties for maintaining your visa or changing visa types if your situation warrants it.