der Stadtteil
district
A defined area or district within a larger city, often with its own character.
Kreuzberg ist ein bekannter und multikultureller Stadtteil von Berlin.
Kreuzberg is a well-known and multicultural district of Berlin.

district
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district
A defined area or district within a larger city, often with its own character.
Kreuzberg ist ein bekannter und multikultureller Stadtteil von Berlin.
Kreuzberg is a well-known and multicultural district of Berlin.

This word is built simply: it consists of 'die Stadt' (city) and 'der Teil' (part). A Stadtteil is therefore literally a part of a city.
Dieser Stadtteil ist für seine vielen Parks bekannt.
This district is known for its many parks.
To describe a place within a district, the preposition 'in' with the dative is almost always used. You live, work, or are 'in einem Stadtteil' (in a district).
Ich wohne in einem sehr ruhigen Stadtteil am Rande der Stadt.
I live in a very quiet district on the edge of the city.
'Stadtteil' (district) is often combined with adjectives that describe its character. Typical examples are 'ein belebter' (a lively), 'ein grüner' (a green), 'ein moderner' (a modern), or 'ein historischer' (a historic) Stadtteil.
Sie suchen eine Wohnung in einem zentralen Stadtteil.
They are looking for an apartment in a central district.
quarter / district
'das Viertel' and 'der Stadtteil' can both mean an area within a city. 'Stadtteil' is usually neutral and often the official or clear term for a part of a city. 'Viertel' sounds a little freer and often emphasizes the character of an area, for example 'ein altes Viertel' (an old district) or 'ein ruhiges Viertel' (a quiet district). The words are therefore easy to confuse. If you simply mean the named part of a city, 'Stadtteil' is very suitable. If you are talking more about the atmosphere or the area, 'Viertel' often sounds more natural.

Wegen der Baustelle fährt der Bus heute nicht durch unseren Stadtteil.
Because of the construction work, the bus is not going through our district today.
In diesem Viertel gibt es viele kleine Restaurants und alte Häuser.
In this district, there are many small restaurants and old houses.