city center; downtown; heart of a city
Often refers specifically to central Tokyo, but can be used for other major cities. Common in news and formal contexts.
都心では家賃が高い。
Rent is high in the city center.
The population of the downtown area is increasing.
He lives in the city center because of his job.
On weekends I go into the city center to shop.
Since rent is expensive, I moved to a place away from the city center.
This apartment is convenient because it's close to the city center.
都心 specifically means the very center of a city (downtown), while 都市部 covers the entire urban area, including areas beyond the center.