First a capsule history: China used to by run by an emperor. The emperor ceded Taiwan to Japan. Then the Nationalists (KMT) overthrew the emperor and established the Republic of China government. The Communists started a civil war to overthrow the Nationalists to make China communist. Meanwhile Japan was invading parts of eastern China. After WWII, Japan abandoned their claim to Chinese territories, including Taiwan. The civil war then picked up steam with the communists eventually capturing from the Nationalists all territories except Taiwan which continues to be administered by the Republic of China government, while the communist territory is administered by the People's Republic of China government. Each side has maintained a general policy that their government is the rightful government of all Chinese territories, while various countries have established diplomatic relations with one side or the other, and maintain a 'One China' policy. 'One China' policy is an intentionally vague concept which various sides try to interpret as meaning different things at different times. It's hard to write all this up in an objective way, but I think that comes pretty close.
The terminology I prefer to use is to refer to the People's Republic of China or PRC when referring to the communist government ruling the mainland, and Republic of China or ROC when referring to the government ruling Taiwan. When referring to one or the other geographically, I use the terms China for the PRC bits and Taiwan for the ROC bits. While somewhat inaccurate, that's the terminology that most people are familiar with. Sometimes though, I have to use the term 'Chinese Taipei' because that's one of the few terms the PRC finds acceptable for referring to Taiwan in international groups it participates in, and will refuse to participate when groups use other terminology.
For my mailing address, I usually use "Taiwan ROC" as the country. This combines the form most people are familiar with and the abbreviation for the proper name of the governmental entity. Alternatively, "Taiwan" or "Taiwan Republic of China" would probably work fine, though spelling out "China" like that will confuse some people, as would just writing "Republic of China".
Because this is all very confusing, and because the PRC often objects when they see certain references to Taiwan or the Republic of China, there's often some very creative ways people have come up with to refer to Taiwan in their ordering systems. It's common to see such things as "Taiwan, Province of China", "Taiwan, China", or "Chinese Taipei" in some ordering systems. These forms are variously confusing, inaccurate, and in some cases intentionally demeaning. And they can all lead to mail going astray.
So back in the middle of January, I decided to order a subscription to Fantasy & Science Fiction magazine. With most subscriptions, I have them sent to my US address and pick them up on my occasional trips. The difference in cost for F&SF was not that much, so I decided to have it sent directly here. I don't remember what they had in place back then, but currently they have a free-form box to enter the country yourself, so chances are I put in "Taiwan ROC" or maybe "Taiwan". I got an email confirmation that said to allow up to 12 weeks for International subscriptions to start, so I waited until a few days ago and then shot off an email to them asking what's up.
The response was sadly that they have invented yet another broken way to write my address, and were sending my issues to "PEOPLES REPUBLIC OF CHINA". And even more sadly, though I'd included my correct mailing address in my email to them, they only noticed that the postal code was different. (We just expanded from three digit to five digit postal codes, and both old and new still work, just like both 5 and 9 digit zip codes in the US still both work.) My reply was quite restrained, given that "Wrong country, DUMBASS!" would probably not be too far out of line. I will make sure they extend my subscription as well as soon as they sort it out.