More precisely, in Japanese, an "American joke" (pronounced ah-may-ri-kan joh-ku using katakana English) is any joke where the intended humor relies on sarcasm, irony, black humor, or absurdist humor, because these are types of humor that generally speaking do not exist in Japanese.
In a Japanese context, the three main forms of humor are slapstick (people taking pratfalls), insult humor (calling people fat or dumb, for example), or puns.
When a Japanese person calls something an "American joke," it does not necessarily mean that they did not get the joke or that it wasn't funny - some Japanese people understand these jokes - but it means that it is a type of joke a Japanese person would not normally make.