Female genital cutting (FGC), otherwise known as female genital mutilation or female genital circumcision, is the ritual removal of part or all of the vulva. It's practiced in regions of Africa and the Middle East, even though it's illegal in most countries. It tends to be most common in areas of greater gender inequality, where women have no options but to be married. The procedure is required to be eligible for marriage and is considered a rite of passage.
The theory behind FGC is that women will be overcome with desire and will have sex before marriage if their genitals are left intact. FGC is intended to maintain their purity by removing any structures that could bring them sexual pleasure, thus reducing temptation.
FGC is usually done around puberty, and in some places at much younger ages. Usually female village elders will do the procedure, and rarely ever is it performed in sanitary and safe conditions (by western surgical standards).
Gena (thanks!) passed along this TEDx clip of a talk by a woman struggling with her cultural roots as an immigrant growing up in Australia, in relation to her beliefs about FGC. Check it out.