A curtsey is a feminine version of a formal bow, a head-dipping, knee-bending gesture of greeting. A woman might curtsey when meeting a Queen.
You show respect when you curtsey, but it's so formal that it isn't often necessary in most cultures. When a woman meets a member of royalty — like a prince or an empress — it's appropriate for her to curtsey. Figure skaters and some dancers also do a ceremonial curtsey at the end of a performance. A more common way to spell it is curtsy — both versions come from the root word courtesy, which you show when you curtsey.