Answer by Makiko Itoh, 海外組:
Oiran (花魁) was a name given to a prostitute who was very popular and highly regarded, mostly for her beauty, in the brothels of Yoshiwara in Edo (Tokyo). In the Edo period, prostitution conducted in specified areas, called yuukaku (遊郭), like Yoshiwara and was legal. A regular Yoshiwara prostitute was called a yuujo (遊女) which means "play woman." (Other types of prostitutes had other names.) An oiran was like the pinup girl of Edo—many of the bijin-ga (pictures of beautiful women) that exist as woodcut prints are of oiran.
There are no oiran left in modern Japan since prostitution is illegal now. There are some borderline almost prostitution businesses around, but the women who work in them are not called oiran or yuujo.
A geisha (芸者), geiko (芸子), or geiki (芸妓), is a trained entertainer who is very skilled in song, dance, playing an instrument, and otherwise entertaining guests. Although not primarily prostitutes, some did sleep with clients and many "successfully" retired by becoming the mistress of a client, or sponsored by one or more, and so forth. There were also male geisha in the Edo period.
A tayuu (太夫) was the name used especially in Kyoto and Osaka for highly lauded and popular geiko and yuujo. Early on, the term was used for similarly ranked women in Edo too, but oiran became more widely used. (The term tayuu is also used in many other contexts for men and women. Basically it does mean the best or most respected in a certain field, such as in noh theater.)
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