Which are the best districts/areas for a hotel in Berlin?

anywhere around Friedrichstraße will get you easily to most sights on foot. The Melia Berlin on Friedrichstrasses near Spree river  is a good location-on line can often get very good rates-only a short walk to trains, Museums, Brandenburg Gate, Checkpoint Charlie, etc. Lots of cheap eating round there plus a wonderful bookstore just down the street. If you're looking at hostels, City Hostel is well located. If you're there a while, an apartment may be an idea-Mondrian apartments and Damn apartments were well located (haven't stayed myself but friends found them good).

My pick is to stay near Savignyplatz (Berlin). The neighborhood is central, historic, has lot's of restaurants, design shops, and cafes. It is next to an S-Bahn line that gets you to the Zoologischer Garten, Potsdamer Platz, Hackescher Markt, Alexanderplatz, among other places.

Recently I stayed at Hotelodeo GmbH, just off Savignyplatz, and recommend it. Nice and really helpful staff. Plus the food and decor are sublime and so well done. The hotel room was very comfortable in furnishings and space. Love this place! Also, they have a lobby with a library of fabulous design, travel, cooking, literary, architecture and other books.

I'd say Mitte or towards Kreuzberg (which is a hip neighborhood nowadays I should add). I've stayed in Hotel NH Berlin Mitte and I found the location very convenient (I don't know if the particular hotel is within your budget but I suggest is so as to point out the location)!

We stayed in Alexanderplatz and it was perfect! 

The Mitte.  "Mitte encompasses Berlin's historic core and includes some of the most important tourist sites of Berlin like Museum Island Brandenburg Gate, Unter den Linden, Potsdamer Platz,  Bundestag"

If you want to be in the middle of more traditional, Unter den Linden.  If you want more modern, more action, Potsdamer Platz.

If you're young and want to party, too: Kreuzberg, Neukölln, Friedrichshain.