A gist on what public and private health insurances are:
Public / Statutory insurance is regulated by the German Government. Almost 90% of the population (73,3 million) contributes to the public health insurance.
Public health insurance also includes cover for your long-term care insurance (Pflegeversicherung).
The costs of public health insurance change with your salary, but are capped at a maximum.
The costs (see below) are split between you and your employer equally.
Private Health Insurance, on the other hand, requires you to qualify under certain conditions. Example: As a salaried employee, you must make more than 64.350 € annually.
Applicants can also be rejected because of previous illnesses or because of their age.
In contrast to public health insurance, the premiums in private health insurance, and the contributions depend on your age (at registration), individual risk of illnesses and disability and deductibles.