Female circumcision is a hot topic in todays contemporary Islamic and non-Islamic worlds. For one the anti-Islamic fanatics consider all kinds of circumcision, even male circumcision, an act of violence against human and a form of child abuse. On the other hand, ignorant traditionalists who don't know much about religion consider it strictly obligatory.
Before we get into the details of female circumcision, we should note that male circumcision is a widely known Islamic practice. The act is not prescribed in the Quran but there are various authentic traditions that it was performed by Prophet Muhammad(SAAW) and he himself was circumcised. And it’s well known that the benefits of male circumcision outweigh the risks. But that’s not what the OP is asking here.
Female circumcision is a pre-Islamic tradition. It was practiced by non-Muslims in pre-Islamic Arabia and other parts of the world before Islam. Only recently it became illegal in the United States 1.
19th century gynaecologists in England and the United States would perform clitoridectomies to treat various psychological symptom as well as "masturbation and nymphomania."
In the United States it became illegal in 1997, and in the same year the WHO issued a joint statement with the United Nations Children’s Fund (UNICEF) and the United Nations Population Fund (UNFPA) against the practice. FGM is a crime in many countries now.
Back to the Islamic view on female circumcision. There’s nothing in the Quran and Ahadith that enforces female circumcision. However, there’s one tradition(Hadith) of doubtful authenticity that permits but does not encourage the practice.
A woman used to perform circumcision in Medina [Madîna]. The Prophet (peace be upon him) said to her: 'Do not cut severely as that is better for a woman and more desirable for a husband.'–Sunan Abu Dawûd, Book 41, #5251.
Before we move on, it’s important to note that female circumcision is not the same as female genital mutilation (FGM). According to the mentioned Hadith, Islam, if at all, permits the most basic kind of circumcision, which is the removal of the genital prepuce– similar to the prepuce of a penis 2. Islam protects the rights of women to sexual enjoyment and health and if the act of circumcision by any chance violates these rights it would be considered Haram. That’s why it’s allowed for a woman to ask for a divorce if her husband cannot satisfy her sexual needs and desires.
You might argue that, well, it is permissible– at least there’s one tradition, no matter how accurate, that says so. To answer that, Islam didn't change everything when it first emerged. Changes were gradual. For example slavery which was a free Islamic tradition wasn't abolished with the advent of Islam. Changes were made in order to improve the handling of salves 3. So we can say that even if Islam did permit female circumcision it was to tolerate the practices of pre-Islamic law.
That said, it might be true that in certain parts of the world FGM is still imposed by parents. But that’s because in this world, old traditions are so much rooted in local cultures that people often confuse them with religious requirements.