And in which context are used. Why are so many words for city in japanese T.T
- 都市 refers to cities in the sense of metropolitan/urban areas as opposed to rural areas. It's also the umbrella term for large municipalities in any country.
- 都会 is roughly the same as 都市 in the sense of metropolitan areas, but sounds more colloquial and less technical. In conversation you will hear 都会に住みたい more often than 都市に住みたい.
- 市 is one of the categories of municipalities, and it corresponds to "(XYZ) city" in English. Its friends are "(XYZ) village" = 村, "(XYZ) town" = 町, etc. For example, The New York City is ニューヨーク市. See: Cities of Japan
- 市街 is more like "(main) urban district" or "downtown" in a city.
For example, special wards of Tokyo are not technically 市. But people living in these areas are safely called 都会人 ("city people") although there is no kanji 市 in their addresses.
And as you can see, there are many words also in English (metropolitan, urban, city, ...) :D