This is self-descriptive. This is when a person who does not originally have an English Language name gives himself or herself one. This is a common phenomenon for entertainers in Hong Kong and to a lesser extent China and Taiwan. Here are some examples, with the actor/singer's name given first in English and second in Mandarin:
Also most H.K.pop singers have English names.(e.g. Jacky Cheung, Andy Lau, Aaron Kwok, etc.) In my mind, the use of English names came mainly from Hong Kong originally due to the English influence.
On the other hand, not everybody has English names, Gong Li and Chow Yun-fat are good examples.
Lots of people in China who deal with foreign companies also adopt English names. In my former job in Shanghai, everybody had English names. All my friends had English names.
I have no conclusive proof, but I suspect that Andy is the most popular "English name" judging by the frequency that I hear people using it.