The country's official name is Zhong Hua Ren Min Gong He Guo--in modern Pin-Yin romanization, I'm not going to try to write it in the old Wade-Giles system. Our English name, People's Republic of China, is a reasonable translation of those words.
In less formal speech, Chinese people call the country Zhong Guo, which means "the central country," or as it's often translated a little more poetically, "the middle kingdom." This refers to the fact that for most of China's existence, it really was the center of that part of the world. "Zhong Guo" has been used for two millennia, and "Zhong" by itself is still widely used on signs and in newspaper headlines to mean simply "China" or "Chinese."
However, our name "China" is, in fact, not a name that the Chinese people have ever used for their country. It's a reference to the Qin Dynasty (which was spelled "Chin" until a few decades ago), which united the various provinces into a single political entity around 200BCE, creating what became the largest nation on Earth for a couple of thousand years. When Westerners began to travel to China and take information back to Europe, they picked up the name "Chin," and before long the country was being called "China" by people who had never seen it.
When the royal government was overthrown in the early 20th century, the new democratic country called itself Zhong Hua Ming Guo, "Central People's Republican Country." When the communists overthrew that government, they added several words to the name that imply a higher level of democracy, and which also harken back to the battles between the republicans and the communists. "Gong He," which we Americans usually spell and pronounce "gung ho," was an old battle cry to rouse the republicans.