Question: What is a "Danna"? It was on Memoirs of a Geisha.? 

Answer: Danna is patron -- they may have certain exclusive rights, are expected to provide assistance with the geisha's laving expenses, help pay off debts,  or provide gifts such as kimono.  They still pay the geisha's regular hourly rate whenever he spends time with her.
This question from the Yahoo Q&A forum is categorized in the topic: Health