China was a country that existed on Earth. It was also a term for a large geographical area in Asia. China bordered against several other Earth nations, among them Russia, Mongolia, Kazakhstan, and Korea. The people and culture specific to China was referred to as Chinese.
While visiting 1986 Earth, James T. Kirk described to Gillian Taylor why he was so interested in George and Gracie. When Kirk asked Taylor if she wanted the details, she told him, "I wouldn't miss this for all the tea in China." (Star Trek IV: The Voyage Home)
During San Francisco's Bell Riots, in 2024, Biddle Coleridge commented on the television coverage of the hostage situation, stating that "great, we're on every channel. I bet they're watching this in China." (DS9: "Past Tense, Part II")
In 2365, Picard, commenting on the hitherto indeterminate location of Iconia, reminded Wesley Crusher that China, too, had been thought a myth until Marco Polo reported his own journey there. (TNG: "Contagion")
Background information Edit
Early drafts of Star Trek: First Contact were to have said that China was a leading member of the Eastern Coalition in the Third World War, but it was decided that politics didn't really matter to the story, so the lines were cut.