The official stuff:

Berlin now (2008) is made up of 12 official districts or boroughs with names a foreigner has no chance to pronounce:
Charlottenburg-Wilmersdorf, Kreuzberg-Friedrichshain, Lichtenberg, Marzahn-Hellersdorf, Mitte, Neukölln, Pankow, Reinickendorf, Spandau, Steglitz-Zehlendorf, Tempelhof-Schöneberg and Treptow-Köpenick.

Just try to say "Treptow-Köpenick". Even some Germans can handle that.

This verbal nonsense is not about neighbourhoods at all, it's just politicians to cut down administrative costs.

Each borough is made up of several localities (Ortsteile in German, sometimes called subdistricts in English). These 96 localities typically have a historical identity as former independent cities, villages, or rural municipalities that were united in 1920 as part of the Greater Berlin Act, forming the basis for the present-day city and state. The localities do not have their own governmental bodies, but are recognized by the city and the boroughs for planning and statistical purposes.

Berliners identify more with the locality they live in than with the borough that governs them.

When Greater Berlin was established in 1920, the city was organized into 20 boroughs, most of which were named after their largest component locality, often a former city or municipality; others, such as Kreuzberg and Prenzlauer Berg, were named for geographic features, in both cases a mountain (in German "Berg"), which is a rare thing in Berlin. Don't take the word mountain too seriously, that's just Berlin humor. The "mountains" are small hills, not even a hundred meters high.