Dumplings are universal. To the best of my knowledge, every culture has some food item which consists of a starchy outside stuffed with filling. If we adopt a broad enough definition, everything from samosas and empanadas to ravioli and pierogi are types of dumplings. Needless to say, China has a ton of menu items that defy translation more specific than “dumpling.”
饺子 (jiao3zi) and 包子 (bao1zi) make up the core concept, though I consider jiaozi to closer to the Platonic ideal of a dumpling. 包, as we covered before, is a general term referring to the wrapping, while 饺 is a specific character for the food item. Jiaozi tend to be sorted by their method of cooking and filling (a.k.a 馅儿, “xian4r” covered here). Boiled (水饺 shui3jiao3, literally “water dumpling”), pan-fried (煎饺 jian1jiao3), or steamed (蒸饺 zheng1jiao3). Pork is the most common stuffing, especially paired with a single vegetable, though one can also get pure beef or lamb stuffed dumplings.
Jiaozi have thin skins while baozi are bread-y, but that distinction gets confused by regional specialties like 小笼包 (xiao3 long2 bao1 “small basket dumpling) from the lower reaches of the Yangtze (Nanjing through Shanghai) and 薄皮包子 (bao2pi2 bao1zi “thin skin dumpling”) from Xinjiang. Of course, we cannot forget wontons (rendered 馄饨 hun2tun), which are always served in soup and have extra skin hanging off them.
Dumplings in Korea and Japan are very similar to the prototypical 饺子, though Japan tends to fry their dumplings more often than China. Linguistically, Japan uses the kanji-equivalent of 饺子 (pronounced gyoza in Japanese). The relation is quite obvious. Korean, on the other hand, uses 만두 (mandu) as its general word for dumplings. This is also related to Chinese, except it derives from 馒头 (man2tou, a steamed bun). This is ironic because mantou are best described as baozi without any filling, which would exclude them from the dumpling family. Finally, I don’t know how widespread it is, but the fried dumplings I ordered (pictured above) had “야끼 교자” provided as explanatory text. Sounding that out, “Yakki Gyoja” sounds just like a transliteration from the Japanese (yaki being that super common element of Japanese grilling cuisine names: yakitori, teriyaki, teppanyaki, okonomiyaki, etc.).