For medium term flat rentals (1-2 months) you could look to Airbnb which works in about any place by now.
Probably cheaper would be if you can find a private place as a temporary short-term rental ( Zwischenmiete in German) at WG-gesucht which is typically young people renting out their places while away for internships/vacations/... Now be aware that you are on the German housing market so some knowledge of German will come in handy even if Berlin is a fairly international place.
Do insist on seeing the place at least via Skype, signing a contract (in German likely) and be careful about scams.
travel.stackexchange.com/users/32134/mts
An alternative to Airbnb: tripping.com ; they aggregate search results from a lot of vacation rentals websites.
Disclaimer: not affiliated in any way to that site, just happened to use it a few times in similar situations as the OP's, with good results (cheaper than Airbnb for comparable listings)
travel.stackexchange.com/users/53646/user4867444
I've lived in House of Nations Studentenapartments Berlin for a short time.
The quality of the housing is just average, but you avoid the possible hassles with finding and renting a room from a private individual on WG-Gesucht (which is otherwise a good choice, and typically cheaper) and you can do more or less everything in English.
Check also CentralHome – First Choice Berlin .
travel.stackexchange.com/users/7835/Federico Poloni
I have rented a few times in Berlin and other German cities through REM
e.
V., Ring Europäischer Mitwohnzentralen .
It was always the case of someone going overseas for a few months/years and looking for someone to cover their rent while they were gone.
I found the people there to be friendly, helpful and efficient.
travel.stackexchange.com/users/3390/Mawg says reinstate Monica
Regarding Berlin in particular, it is often illegal to turn an apartment with long-term rental (housing) into one with vacation rental (tourism).
As a rough approximation, it would be a violation of zoning laws (Zweckentfremdungsverbot).
Politics and law enforcement have been slow to catch up with the massive growth of online portals, but they are starting to do so now.
The legal problems will be for the landlord, not for you, but I would not rely on the likes of AirBnB until that is settled.
Also, you will have to register with the municipal authorities as a resident (after arrival), that may be difficult without a proper rental contract.
In all likelihood the Bürgeramt will only process your registration after you have left, but you have to go through the motions anyway.
travel.stackexchange.com/users/27778/o.m.
"I understand that your question is about renting an apartment, but why don’t you ask somebody to “borrow you the keys” from his apartment?!
Last summer I visited Spain with my friends, one of them had found a good apartment for us in Barcelona for free.
We lived there for about a month and were very thankful to the owner for his fearlessness and trust;)
I learned that my friend had found a new service of home exchange or house-swap, easy to register and use, and we stayed for free due to this awesome service.
Thus I recommend to find a family that can leave their flat for you here
http://swap-house.com/host/germany/
I hope it will be useful.
Good luck"
travel.stackexchange.com/users/53761/Emily Lane
Next summer I will be attending a two-month course in Berlin with my friend. We are looking for a small flat (studio/one bedroom) at a reasonable price (ideally  in the Mitte district).   Where can I look for it or who may I contact?
Flat renting in Berlin for medium-term period
travel.stackexchange.com
<germany><accommodation><rental><berlin>
travel.stackexchange.com/questions/82534
travel.stackexchange.com/users/53636/Paulina
You can find medium-term flat rentals on Airbnb or WG-gesucht, which is often apartments rented by young people while away for internships/vacations. Tripping.com also aggregates search results from many vacation rental websites. CentralHome – First Choice Berlin is also worth checking. Airbnb appears to be the best option for a medium term rental in Berlin. Other options include CentralHome - First Choice Berlin and House of Nations Studentapartments.
Airbnb and short-term rentals seem to be the options for a two-month stay in Berlin. House of Nations Studentapartments and CentralHome-First Choice Berlin are alternatives to Airbnb.