Technically there is a
difference between A British High Commission and A British Embassy.
As
D4109125 advised, it doesn't really matter what you write in this but if you want to be précised, I would suggest write something like
British Embassy Beijing if you are talking about Beijing in China (just an example).
The word 'High Commission' is only used for Common Wealth Countries. For all other countries, the word 'Embassy' is used, for example
British Embassy Paris.
As China is not part of the commonwealth, it will have to be a British Embassy
*, China (where * is the relevant City in China). The commonwealth is a group of countries that are colonised by Britain and give sovereignty to the Queen of England. China has never been colonised by Britain. Hong Kong has, but it was relinquished in 1999. The commonwealth games are games for members of the commonwealth only. China participates in the Asian games which is the Asian equivalent of the commonwealth games.