Dynasty means a succession of rulers, all from the same family.
In monarchies, historians can describe different dynasties:
- In the England there were the Plantaganet, Tudor, Stuart, and Hanover dynasties.
- In Ancient Egypt the dynasties are numbered from 1 to more than 30.
- In China the dynasties are named after the clan name of the ruling kings
In China in particular, "dynasty" is used as a translation of Chinese "朝".
In general, a dynasty isn't the name of the country, but may be used as a way of describing a certain period of a country or empire, and the culture of the country at that time. The name of the country is "China" (中国)
So we would say that Liu Bang, of the Han clan, became Emperor of China, in English.