What does "comprehensive health insurance" mean?
If you want to study in Germany or are staying in Germany for a longer period of time for other reasons, you need health insurance. This is mandatory and protects you from high medical costs that you would otherwise have to pay yourself.
You can choose between statutory and private health insurance. Statutory insurance offers you good basic coverage. It covers the costs if you need to see a doctor, dentist, or go to the hospital.
What exactly statutory health insurance pays for is regulated by law. There is a catalog of benefits ("Leistungskatalog") that all statutory health insurance organsiations must adhere to.
You can see that everything is legally defined in statutory insurance. That's a good thing, because it means that everyone living in Germany is entitled to the same protection and no one is disadvantaged.
However, if you want a little more freedom in your insurance coverage, private health insurance is better for you.
Which insurance policies are NOT comprehensive health insurance?
Make sure that it really is comprehensive insurance. There are insurance policies that are only a supplement to statutory health insurance. These are called “supplementary health insurance.”
You can take out this insurance if you have statutory health insurance and would like to receive a little more reimbursement for dental treatment, for example. Supplementary insurance can also be useful if you need glasses. Health insurance only covers part of the costs in these areas.
Also standard travel health insurance is not comprehensive health insurance. You will often discover this when you research the topic. Travel health insurance is only intended for vacation trips to another country, not for longer stays.
In order for a university to recognize your health insurance, you must provide proof of comprehensive health insurance.