der Bezirk
district
An administrative area within a larger city or region with its own administration.
Berlin ist in zwölf Bezirke unterteilt, wie zum Beispiel Mitte oder Pankow.
Berlin is divided into twelve districts, such as Mitte or Pankow.

district
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district
An administrative area within a larger city or region with its own administration.
Berlin ist in zwölf Bezirke unterteilt, wie zum Beispiel Mitte oder Pankow.
Berlin is divided into twelve districts, such as Mitte or Pankow.

To describe a place within a district, the preposition 'in' is used with the dative. The question for this is 'Wo?' (Where?). For example: 'im Bezirk' (in the district).
In diesem Bezirk gibt es viele internationale Restaurants.
There are many international restaurants in this district.
In an administrative context, 'Bezirk' (district) is often combined with verbs like 'wohnen' (to live), 'arbeiten' (to work), or 'verwalten' (to administer). Belonging to a city is also often mentioned.
Er wohnt im Bezirk Neukölln.
He lives in the Neukölln district.
When 'Bezirk' (district/area) describes a general area, it is often accompanied by adjectives that clarify the character of the area. Typical examples are 'ein ruhiger Bezirk' (a quiet district), 'ein grüner Bezirk' (a green district), or 'ein historischer Bezirk' (a historic district).
Sie suchen eine Wohnung in einem ruhigen Bezirk am Stadtrand.
They are looking for an apartment in a quiet district on the outskirts of the city.
area / field
'der Bereich' and 'der Bezirk' can both mean a defined area. 'Bereich' is more general and works for almost any zone, for example in a building, in a city, or on a map. 'Bezirk' sounds more specific when talking about part of a city or region, often with administration or official boundaries. So you would be more likely to say 'der ruhige Bereich im Park' (the quiet area in the park), but 'der Bezirk der Stadt' (the district of the city) when an official city district is meant.

Unser Bezirk bekommt nächstes Jahr eine neue Bibliothek.
Our district is getting a new library next year.
Im vorderen Bereich des Bahnhofs ist es morgens besonders voll.
The front area of the train station is especially crowded in the morning.