A borough is usually a metropolitan area [city or town], whereas a district can be either part of a city, or a suburban or rural area. Large English cities are divided in to boroughs - London has 33 - and smaller cities and towns are single boroughs. Districts may include a number of smaller towns.

I see. Thank you for answers.

district and borough are words used to describe areas. In America, there are 50 states. In the 50 states there are many cities. In cities (not all cities only some), there are boroughs. New York City has 5 Borough, Manhattan, Brooklyn, Queens, The Bronx, Staten Island. Many cities do not have boroughs. Districts is a very general term. It can mean a part of any area. It depends on the context. If you are in a city, a district may refer to a part of a city. Sorry it's very confusing!

It depends on what the city government refers to it as - in New York you have 5 boroughs but where I live in Washington D.C it has 4 districts Northwest, Southwest, Northeast and Southeast. I may not be 100% correct, but I think New York is the only city in the United States that uses the term borough - it's either neighborhood or district everywhere else.