Things I know about geisha...

The purpose of geisha is to entertain men (but not necessarily in the way your thinking, although they do sometimes) . They can't do things like serve food to their guest but they pour them drinks and chat and sing and dance and play the shamisen and accompany them around.
They mostly live in houses called Okiya, where there are maids, a mother(the owner) and sometimes Aunties, and geisha and maiko(trainee geisha). These people are not necessarily blood related.
There are also youner and older sisters. When a girl becomes a maiko she is appointed an older sister who is already a geisha, to teach her and introduce her to customers and the owners of tea houses. Teahouses are where geisha party's are held.
To become part of an okiya or to be appointed an older sister you must take part in a ceremony, usually involving tea or warm sake (umm).
Tea ceremony is very important to a geisha.
Most geisha try to have a patron, or danna, who is a man who has bid for the geisha and will then keep her in kimono and expensive gifts and a good quality of live. All geisha want a danna (preferably a nice one) because the cost of lessons in the geisha arts can be very high, and all geisha are in debt to the okiya for the investment it put into her as a maiko.
A man can also bid for a geisha's mizu-age (virginity). After this the geisha is not tied to her mizu-age patron in any way.
Original geisha had no say in who their danna or mizu-age patron was.
Geisha used to be the trend setters of Japan, but as times changed they found it hard because of their traditional look and style of art. They did try to become more modern but it wasn't very successful. They are now recognized as one of the most famous and traditional things about Japan.
Of course geisha of today are very different from the Edo period in 1600.
The Gei in geisha means Art.