The most well known insurance companies are AOK and
TK
You can contact them to ask which service they provide
and a contract that you will complete and send.
You will receive then your Health Card that you must
sign. You should show it every time that you see a doctor
and then you won't have to pay.
The card holds the following information: your name,
your surname, your birth of date, address, insurance
number and two insurance companies numbers. No medical
data is stored. The card is valid
for 6 months maximum.
The card is not valid aboard. In foreign countries
when there is a security social agreement, you can use
the European form E111 or Auslandskrankenscheine. This
form must be completed by the foreign doctor to receive
the German reimbursement of the fees. When you leave
Germany, always use that form.