Your visa will indicate how many times you are permitted to enter China under that visa. On the first line, there should be a field "ENTRIES" with a letter and a chinese character following it. If that letter is M, you are eligible for multiple entries. China also has single and double-entry visas; presumably, those would be the letters S and D respectively.
If you have a double-entry visa and you have only visited China once or if you have a multiple entry visa you are eligible to re-enter. However, if you have a single-entry visa or have used both entries of a double-entry visa, you must apply for a new visa from the Chinese consulate.
Hong Kong, while under Chinese sovereignty, is run by its own government and has its own immigration controls. US citizens are eligible to visit Hong Kong for tourist purposes for up to 90 days without a visa so long as you have a departure ticket and adequate funds.
Visiting Tibet requires documentation additional to the Chinese visa, namely the Tibet Travel Visa and a PSB permit. These permits are usually granted to visitors who are on a pre-arranged group tour, and China does close Tibet to outside visitors from time to time.
Taiwan is de facto its own country and has its own immigration controls. US citizens are eligible to visit Taiwan for tourist purposes for up to 90 days without a visa so long as you have a departure ticket and adequate funds.
