What do the Chinese people call themselves?

Chinese and other people from that part of Asia are called Asians

Chinese call themselves "renmin" ("people") in Chinese. The Chinese spoken language doesn't make much of a distinction between "r" and "l", which is where all the "flied lice" jokes come from. So if you say "renn-meen" you'll sound like an English speaker, but will still be understood without any problem.

China itself is "zhong-guo" ("middle country", pron. approx. "chunn-gwo")

As is true of English, but not many other languages, Word order is very important in Chinese. We see that in English when we realise that "Jane hit the ball" means something completely different to "the ball hit Jane" even though all the Words are the same, just in a different order. Chinese is like that too, which is part of the reason why it's easy for us to understand "long time no see", which is a Word-for-Word translation from Chinese.

So, the Chinese term for a Chinese person is "zhongguo ren" - "China person" or, more exactly, "middle-country person"

Chinese also call themselves the Han people or Tang people. This is especially the case for overseas Chinese. Han and Tang are two high points in the Chinese history.