China as a country has had many names over the centuries it has existed. Some names have stuck, some have not. It is fun while looking at history to consider some of these.
Zhong Guo. One of the longest endearing and perhaps most profound name for China is its Chinese name of Zhong Guo or "middle kingdom". Historically, it did literally mean the "central" kingdom during the period where China was very fragmented with a northern, western, and southern kingdoms as well. However, over time, this name was adopted to take on more political implications. In simplest terms, whereas the old Western world consider the world flat, China considered itself the center (or middle) of the universe. All things revolved around Zhong Guo and there was great pride in having one country.
Canton. Early on in China's interactions with the West, it was not uncommon to see the areas of China referred to as "Canton". This area roughly translates to today's Guangdong Province in Southeastern China, where a lot of the trade occurred. The word "Canton" originally was how this area's name was originally represented in Portuguese or French. As other Westerners interacted with the area, the name was adopted to represent the land as a whole. Interestingly enough, this word still makes its presence today as describing the dialect that is derived from this region, Cantonese.
Cathay. This is another historical name which gained momentum as the Western world interacted with China. Generally, this name refers to group that dominated Northern China during the time that Marco Polo traveled this land. There are many references from his literature of the Manchurian kingdom, from which these people originated.
People's Republic of China and Republic of China. At the end of the Chinese Civil War (which roughly coincides with the end of World War II), mainland China adopted communist rule under Mao Zedong. It was at this time that the People's Republic of China was established. At a similar time, the remnants of the previous government, Republic of China (which existed since the end of the imperial rule roughly at the turn of the century), established themselves in Taiwan to create a "second China".
China. Even this very modern and general name for the land has a rich history. Its origins seem to derive from Sanskrit literature "chin", which either refers to a specific tribe or a way to write the name of the Qin Dynasty.
There are other names and references to China which perhaps have less "historical" settings.
Zhong Hua. Another common name that is used for China can be translated into the middle of prosperity or grace. This is meant to be an uplifting way to refer to Zhong Guo, which has a more political and military feel to it. Personally, this is one of my favorite ones as my middle name is that same "hua".
The Rooster. Finally, one more comical reference to China that many Chinese make is to call the land "the Rooster". If you look at mainland China on a map, you will notice that it somewhat looks like a rooster. The head is to the far northeast, the tail feathers to the northwest, and the two feet are the islands along its coast.
There are many more names that China has been known through time. Many have come and gone based on the various changes and shifts that the land has gone through over the centuries. All point to the vast richness of history that exists within this country and its lands.