das Viertel
district
A part of a city; a neighborhood or a specific district.
Wir wohnen in einem sehr lebendigen Viertel mit vielen Cafés.
We live in a very lively district with lots of cafés.

quarter / district
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district
A part of a city; a neighborhood or a specific district.
Wir wohnen in einem sehr lebendigen Viertel mit vielen Cafés.
We live in a very lively district with lots of cafés.

The word 'Viertel' (quarter) is derived directly from the number 'vier' (four). All meanings relate to the idea of a 'fourth part'—whether a part of the city, a quarter of a whole, a quarter of an hour, or a quarter liter.
Für das Rezept brauche ich nur ein Viertel Liter Milch.
For the recipe, I only need a quarter liter of milk.
The context makes the meaning clear: In connection with a city, it refers to a district. With 'nach' (past) or 'vor' (to/before) and a time, it means a quarter of an hour. When ordering wine, it means a quarter liter.
Wir treffen uns um Viertel vor acht.
We're meeting at quarter to eight.
When referring to a district, 'Viertel' (district) is often used with prepositions such as 'in' (in) or 'durch' (through). Typical combinations are 'in einem Viertel wohnen' (to live in a district) or 'durch ein Viertel spazieren' (to stroll through a district).
Ich wohne in einem ruhigen Viertel am Stadtrand.
I live in a quiet district on the outskirts of the city.
For times, 'Viertel' (quarter) is used in fixed expressions: 'Viertel nach' (quarter past) for 15 minutes after the full hour and 'Viertel vor' (quarter to) for 15 minutes before the full hour.
Der Zug fährt um Viertel nach fünf ab.
The train departs at quarter past five.
In parts of eastern Germany and Austria, you often hear a different time expression: 'dreiviertel' (three quarters) followed by the next hour means 'Viertel vor' (quarter to). For example, 'dreiviertel acht' means the same as 'Viertel vor acht' (7:45).
Mein Termin ist um dreiviertel zehn.
My appointment is at quarter to ten.
four
'vier' and 'das Viertel' are both connected to the number 4, but they do not mean the same thing. 'vier' is the number itself: four people, four books, four hours. 'Ein Viertel' is a part of a whole, namely one of four equal parts. Learners may confuse the words because 'Viertel' comes from 'vier'. The important point is: 'vier' counts whole things, while 'ein Viertel' describes a share.

Tom isst nur ein Viertel der Pizza, weil er keinen großen Hunger hat.
Tom eats only a quarter of the pizza because he is not very hungry.
Für den Termin brauchen wir vier Stühle im Besprechungsraum.
For the appointment, we need four chairs in the meeting room.