If you are an employee, your insurance costs will be covered as part of your taxes, and this would be done through the company you are working for. However, it’s usually your responsibility to sign a contract with an insurance company of your choice and provide your insurance number to your employer. Normally, the costs are similar for all insurance providers, so whether you choose AOK, TK, DAK (those are among the most known) or any other, the costs would not be dramatically different.
Depending on employer and insurance company, there is a chance that your family can also use your insurance account as well.
The cost of health insurance is ca. 14% of your gross salary, full tax package (legally required) is around 42%.
In all other cases, you have to get private insurance. Rates vary for different providers, so you have to look for the one which suits you the most.
I hope this helps.