Residencies in Germany are like jobs. Once you have your certificates. You can apply through email or send application by post to HR department (Personal Abteilung) of Hospitals which have vacancies. If they like your CV and experience they will call you for Interview which is obviously conducted in German language. Once you are selected by the Hospital, you will receive an appoinment letter , with that you can apply for Berufserlaubnis ( Temporary German Medical License which is valid only to the particular State). This can take anywhere between 2 weeks to 5 months depending upon the state in which you apply. After receiving the license one can officially start working as a PG trainee and start receiving monthly salary of around 4000 Euros gross, which after taxes shrink to around 2300-2400 Euros /Month.
Berufserlaubnis gives you the time (6 months to 2 years depending upon State) to start working for the time being and get a Approbation (Permanent License) in the stipulated time.
Primary Medical training in Russia and Eastern Europe is sometimes considered equivalent to that in Germany when compared number of hours of teaching in each subject and duration of the course therefore these doctors many times donot need to apply for Berufserlaubnis, they can directly get a Permanent license ( Approbation -valid all over Germany) without any exam.
For countries from Indian subcontinent ,Africa and Gulf, Doctors need to take a exam (Knowledge Test/ Gleichfertigkeit Prufung) inorder to get a Permanent Medical License in Germany.