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.)
More questions on Japanese Culture:
TODAY IN SLATE
Meet the New Bosses
How the Republicans would run the Senate.
The Government Is Giving Millions of Dollars in Electric-Car Subsidies to the Wrong Drivers
Scotland Is Just the Beginning. Expect More Political Earthquakes in Europe.
Photos of the Crowds That Took Over NYC for the People’s Climate March
Friends Was the Last Purely Pleasurable Sitcom
This Whimsical Driverless Car Imagines Transportation in 2059
Did America Get Fat by Drinking Diet Soda?
A high-profile study points the finger at artificial sweeteners.