, literally "name" or even simply "Chinese character
The "courtesy" or "polite name" is the main name by which a Chinese man was known in traditional culture. Some women in good families had courtesy names, too, but it was universal for men. This name was traditionally assigned by a family elder after the young man reached his teens, and from then on it was the main way he was addressed when not being called by his official title (Gentleman Lin, Historian Sima, etc.). His true given name (ming2) would be used only by relatives and old friends - friends and colleagues would call him by his courtesy name, and he would sign it to letters. The courtesy name is a little like the agnomen or added name in ancient Rome, except that every adult man had one.
In the biographies of the Chinese dynastic histories, each person's biography ordinarily begins by telling the subject's birth name, courtesy name, and place of origin (not necessarily the birthplace, but the traditional place of origin of his or her branch of the family!). Because so few surnames are in common use in China, it would not do to refer to a man by surname alone ("Zhang then had an audience with the Emperor...") because there would be hundreds of other people with the same name with whom he might be confused. But a courtesy name was typically very distinctive, bearing elegant literary allusions or rare characters, and there was little confusion among them. Your parents might name you something really ghastly, but your courtesy name had to be something that sounded cultivated. Chinese writers in pre-modern times often wrote about eminent figures of the past using their courtesy names, which makes it sound as if they knew each other - quite strange! Confucius, for instance, is often called Zhong4-ni2 by traditional scholars as recently as the 20th century. This practice reminds me of the way, in 18th century England, ancient Romans were sometimes called familiarly by a modernized form of the nomen or gens name - "Tully" for Marcus Tullius Cicero, for instance.
The practice of choosing courtesy names has gone out in Chinese society, except among artists - who may be forgiven a proclivity for pseudonyms and other masks. (And pseudonyms, like IRC handles, are technically hao4, not zi4.) Nonetheless courtesy names have one clear modern survival. Foreigners who study Chinese usually take a Chinese name in their first year or two of study, and these are very much like traditional courtesy names. Your Chinese name isn't you real name - that will be in English or French or Yoruba or what have you - but it is used whenever you speak Chinese and when Chinese-speaking people talk about you. Close friends may choose to attempt your real foreign name, but everyone else will use your Chinese name (which you will print on your name card, for their ease) unless you tell them otherwise. This modern separation between foreign and Chinese names is very much like the traditional separation between zi4 and ming2.