As others stated, you have the public and the private system. The latter is accessible after you earn above a certain threshold for a certain number of years (no idea about the specifics as I would have been eligible but never bothered to change). You also have niche solutions for certain free professionals (doctors, artists,...).
I would call the system pretty robust, as you cannot be uninsured. From personal experience: Having my kids insured under my name, under my wife's name when I earned less than her, being insured under my wife during a certain time of a sabbatical, insured through unemployment and now as an entrepreneur I never changed insurance provider. Only the status changed.
Interesting detail, even the public health insurance sector is run by commercial insurance companies. The minimum plans are defined by the state. Regardless of how many people are insured under your name, premiums don't change and stay at a certain percentage of your salary. Again, slight differences between providers but usually around 15 or so percent. Currently I pay much less, as there is no revenue yet.
Also nice, premiums don't change with previous sickness and preexisting conditions. Also 6 weeks of paid sick leave per sickness, resets for each sickness after a certain number of months (again, don't ask the exact number).
One important insurance to add is continued salary after the six weeks are over. Costs close to nothing when done early and can save you. Non serious things tend to be over well within 6 weeks. But serious ones can take considerably longer. And the last thing you want is money problems the. I spend 6 months on sick leave once, didn't have to worry about my job nor money. Was a nice thing.