As a Chinese name:
The Jung 鄭 surname comes from Chinese people of the Yin-Ping region of Guang-dong province in China.
Jung is the Cantonese pronunciation of the Chinese family name 鄭 for those immigrants coming to North America in the 50-60s. It
More recently it is also sometimes written as Chang or Cheng in the Hong Kong dialect. In Mandarin the name is often written as Zheng.
As a Korean name:
The "Jung" (Hangul 정) (Hanja 鄭, 丁, 程) is a Korean surname in that currently represents roughly 5% of the South Korean population. It's revised Romanization is Jeong, but can also be seen spelt as Chung, or Cheong. The Korean Jeong surname can originate from three homophonous Chinese characters (or Hanja) 鄭, 丁, and 程.
As a Germanic name:
The Jung surname is also a common Germanic name that is pronounced as "Young" rather than with the English "J" sound.