The basics are included - hassle-free
In public health insurance, all "basic things" (German: Regelversorgung) are covered. You will be sent an insurance card from your public health provider, which you can show to any doctor or any hospital and they will cover you. You don't have to pay anything upfront; all payments to the doctor/hospital will be made directly by the insurance company.
Advanced treatment may cost extra
Public health insurance is good basic cover, but in some cases, you may have to pay for some treatments yourself. Typical things that may not be fully covered are:
Dental treatment, like dental cleanings or tooth replacements (possible to buy supplementary dental insurance, which would cover these)
Going straight to the specialist without a prescription (you must always see a general practitioner first)
Single bedroom in hospital (possible to pay the difference out of pocket)
Co-pay of 10% for medication (up to €10 max per medication)
In this case, the doctor will inform you before the treatment that it will cost extra, which you'll have to pay out of pocket. Following this, you'll have to get in touch with your health provider for partial reimbursement.