Official Terms: 元 角 分

The official currency of the People's Republic of China is the renminbi (人民币), or "People's Money". This word can only be used in the abstract: "Do you have any renminbi?", for example. The British sterling is a similar concept (thanks JudyT).

The official base unit of this currency is the yuan (元), international currency code CNY. All prices in China are denoted in yuan, usually either as ¥5 or 5元; and yes, the ¥ symbol can confusingly also mean the Japanese yen. However, people tend to find the difference between the currency and its unit a little hard to grasp, and it's not uncommon to hear lao wai speak of something costing "5 renminbi" or "10 RMB". Strictly speaking, this doesn't make sense (how much is "5 Chinese money"?), but evidently some people still find "RMB" easier to say than "yuan".

The official subdivisions of the yuan are the jiao (角), at 10 jiao to the yuan, and the fen (分) at 10 fen to the jiao. A coin worth ¥0,10 will thus say 壹角 ("1 jiao"), not "10 fen", on it.

Historical Terms: 圆 圓

To prevent forgery, the yuan's character is formally written 圆 on banknotes and such. Even this is a little simplified: the traditional form is 圓, which shows the shell radical 貝 inside a box and hence harks back to the days when shells were actually used as currency.

The more complex formal versions of the Chinese numerals are also often used for money: 壹 for 1, 伍 for 5, 拾 for 10 and 佰 for 100. These make it impossible to eg. turn a one (一) into a ten (十) just by adding a stroke.

Practical Terms: 块 毛

But in colloquial Mandarin, nobody ever speaks of yuan; the standard term is kuai (块), and the jiao is also dubbed the mao (毛) instead. The fen remains the same, so a price like ¥3,75 would thus be read as "san kuai qi mao wu fen".

The etymology of these alternate forms is obscure. Kuai (traditional form 塊) means "lump" and consists of the rather non-auspicious radicals earth 土 and ghost/devil 鬼, while mao just means fur.

Non-PRC Chinese Currencies

The Republic of China (Taiwan) uses the New Taiwan dollar (TWD), written 圓 or NT$. The Taiwanese dollar is theoretically divided into cents, written with jiao/fen as on the mainland, but this has long since been obsoleted by inflation.

Hong Kong has the Hong Kong dollar (HKD), divided into cents. Interestingly, the jiao is not officially used and eg. coins say "10 fen" instead.

Macau's currency is the pataca (MOP) in Portuguese, divided into 100 avos, but in Chinese it's written using the yuan character (圓) anyway.

And finally, Singapore's dollar (SGD) isn't really a Chinese one, but the locals call and write it as both yuan and kuai anyway.