Removal or splitting of the hood or labia
The term female circumcision is often used to mean several different things. Circumcision literally means "cutting around." In a body modification context female circumcision means removal or partial removal of the clitoral hood. It may also simply mean the splitting of the hood. The term also sometimes covers the removal of the labia minora.
The word is often misused, sometimes to referring to the removal of the clitoris (clitoridectomy). This is not what it means, nor does it mean the removal of the clitoris, hood and labia, and the sewing of the wound (leaving only a small hole for urination/menstruation) as is practiced in some African and Middle Eastern societies. This procedure is called infibulation and is almost always performed on young girls who do not consent.
As far as why, some people may simply not like the appearance of their genitals. Other people may find sex uncomfortable due to overly large labia. Other people modify the hood to increase sexual pleasure by exposing the clitoris. Hood splitting is sometimes done to make it easier to place a clitoral piercing. Generally, women enjoy the extra sensitivity due to having their clitoris exposed, but some find it painful. Most of those will get used to the new sensations (and experience a small loss of sensitivity due to increased contact with clothing), but a few women still find the sensations too intense even months after the procedure.
Plastic surgeons who do genital surgery (look for ones that do penis enlargement and breast implants) will usually do these procedures with no questions asked. There are also a number of private practitioners who keep a low profile for obvious reasons. Ask subincision practitioners, as some are willing to do these procedures. A plastic surgeon will probably perform the procedure with a laser, making it very fast, and with no bleeding. A bodymod practitioner is likely to clamp the area to be removed to stem the blood flow, then remove the tissue with a scalpel or surgical scissors. In both cases, the healing time tends to be very quick.
To prevent the abuse of underage girls (and often not understanding that someone might WANT to do this to themselves), the laws in many countries are very widely drafted, and may prohibit the procedure from being done by anyone but a doctor. In other countries, even a doctor may not perform the procedure unless there is a medical need.