As a student you have a ton of options to choose from. You can go with:
- Public system (for example TK)
- Expat health insurance (for example Hanse Merkur)
- Full private insurance (Speak to an agent)
Here's the thing. If you CAN sign up for the public system, you should. In 99% of cases it's the best thing to do.
The requirements to sign up are simple. You have to be under 30 years old, and be studying for less than 7 years so far.
But you have to commit to the public system at the beginning of your studies (by law). You CANNOT switch partway through your studies.
For more information on this topic, check out this blog post.