zhōng wén – Chinese language. It can cover both the spoken language as well as the written form. Also, all the different Chinese dialects are considered to be zhōng wén, though with the increasing popularity of Mandarin worldwide, zhōng wén tends to refer mainly to Mandarin Chinese.
hàn yǔ – hàn refers to hànzú, or the Han ethnic group. Of the 56 ethnic groups in China, the Han people account for over 90% of the population. hàn yǔ, as the name suggests, refers to the language of the Han ethnic group, or Mandarin. Mandarin Chinese is considered “standard” Chinese — as opposed to other Chinese dialects.
pǔ tōng hua – The literal meaning of pǔ tōng huà is “common language”. pǔ tōng huà is the official language of Mainland China.
zhōng wén is used when the opposite may be English, Japanese, etc., whereas "pǔ tōng huà" is used when the opposite may be other Chinese dialects. Like in most countries, the most standard Chinese dialect can be found on television. In China, all the TV anchors are required to speak standard pǔ tōng huà.
pǔ tōng huà and hàn yǔ are pretty much the same thing. You can think of pǔ tōng huà as the more standard, better-pronounced hàn yǔ.
guó yǔ (国语) – guó yǔ (国语) literally means “national language”. Guó yǔ and pǔ tōng huà are also essentially the same. guó yǔ (国语) is used by Taiwan and sometimes Hong Kong to refer to Mandarin, while pǔ tōng huà is used by Mainland China to refer to Mandarin.
huá yǔ / huá wén – These two terms also refer to standard Mandarin but are used in Southeast Asia, particularly in Singapore and Malaysia. Both countries have adopted Mainland China’s pǔ tōng huà along with simplified characters rather than traditional. huá yǔ / huá wén refer exclusively to spoken and written language.
The differences between zhōng wén (中文), hàn yǔ (汉语), pǔ tōng huà (普通话), guó yǔ (国语), huá yǔ (华语) and huá wén (华文) are pretty subtle and the nuances are commonly overlooked by foreign Chinese learners.