die City
city center
The center of a larger city, especially the area with the most shops, offices, and cultural institutions.
Wir treffen uns um 18 Uhr in der City zum Abendessen.
We're meeting at 6 p.m. in the city center for dinner.

city center
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city center
The center of a larger city, especially the area with the most shops, offices, and cultural institutions.
Wir treffen uns um 18 Uhr in der City zum Abendessen.
We're meeting at 6 p.m. in the city center for dinner.

The word 'City' (city center) is an Anglicism, that is, a loanword from English. In German, it is often used in colloquial speech and informal contexts for the city center.
Lass uns später in der City einen Kaffee trinken gehen.
Let's go have a coffee in the city center later.
Pay attention to the preposition and case: when you are moving into the city center, you say 'in die City' (accusative). When you are already there, you use 'in der City' (dative).
Wir fahren am Abend in die City, um ins Kino zu gehen.
We're going into the city center in the evening to go to the movies.
Common activities associated with 'City' (city center) are 'in der City shoppen' (to shop in the city center), 'in die City fahren' (to go into the city center), or 'in der City bummeln' (to stroll around the city center).
Am Samstag gehe ich mit Freunden in der City shoppen.
On Saturday, I'm going shopping with friends in the city center.
city
'die Stadt' and 'die City' can both refer to the center of a larger city. In this use, for example, people say 'in die Stadt gehen' (to go into town / into the city center) when they go to the downtown area to shop. 'City' also refers to this central area with shops, offices, and often a lot of traffic. The difference is mainly in style: 'City' sounds more modern and somewhat colloquial or advertising-like, while 'Stadt' is more neutral and general. They can be confusing because both are used in everyday language for the city center. In more formal or completely ordinary sentences, 'Stadt' is often more natural.

In der City sind am Samstag alle Parkplätze voll.
In the city center, all the parking spaces are full on Saturday.
Nach dem Arzttermin geht Sofia noch in die Stadt.
After her doctor's appointment, Sofia is still going into town.