When looking for an apartment, the housing office in your town or community can help you with finding accommodation. The office often arranges accommodation directly. If this is not the case, the housing officers can help you by supplying useful addresses and information. you might also inquire in the housing offices in the districts of interest to you. They offer information on "Wohnungsgesellschaften", which are state-run housing associations that provide apartments for rent.
I attach below a listing with the addresses of the housing offices in Berlin according to the different districts. Hope it helps :)