Circumcision is mandatory for Muslims based on tradition, but it is not mentioned in the Qur'an itself.
The following section is found in the standard Shafi'i manual of Islamic law which states in English translation:
BOOK E: PURIFICATIONe4.3 Circumcision is obligatory (O: for both men and women. For men it consists of removing the prepuce from the penis, and for women, removing the prepuce (Ar. bazr) of the clitoris (n: not the clitoris itself, as some mistakenly assert). (A: Hanbalis hold that circumcision of women is not obligatory but sunna, while Hanafis consider it a mere courtesy to the husband.)"
(Source: Reliance of the Traveller, A translation of the classical manual of Islamic Sacred Law (Shari'ah) `Umdat as-Salik by Ahmad ibn Naqib al-Misri (d. 769/1386), in Arabic with facing English text, commentary and appendices edited and translated by Nuh Ha Mim Keller, Revised Edition 1994, p. 59; A: ... comment by Sheikh 'Abd al-Wakil Durubi; Ar. Arabic; n: ... remark by the translator; O: ... excerpt from the commentary of Sheikh 'Umar Barakat)
The Muslim site Islam Q&A gives a general explanation of the Islamic reasons for (male) circumcision
The issue of female circumcision is very contentious. Note first that the above translation of the Manual of Shafi'i jurisprudence (fiqh) provided by Nuh Hah Mim Keller is not correct, since what the Arabic actually says is:
"Circumcision is obligatory (for every male and female) by cutting off the piece of skin on the glans of the penis of the male, but circumcision of the female is by cutting out the clitoris (this is called HufaaD)."
Further discussion, including the Arabic text of this passage, is found in this article.
In the following some relevant quotations from the Muslim traditions will be given:
Narrated Jafar bin 'Amr bin Umaiya:
... When the army aligned for the fight, Siba' came out and said, Is there any (Muslim) to accept my challenge to a duel? Hamza bin 'Abdul Muttalib came out and said, O Siba'. O Ibn Um Anmar, the one who circumcises other ladies! Do you challenge Allah and His Apostle? Then Hamza attacked and killed him, ... (Sahih Al-Bukhari, Volume 5, Book 59, Number 399)
Apparently, one who circumcizes other women was used as insult against a non-Muslim by Muslim. This suggests that the practice of circumcizing women was widespread among the pagan Arabs before Islam, and the lady who performed these kinds of surgery was seemingly not considered a very respected person in society, or it would not be possible to use this as an insult.
That this occurred also among Muslims is shown by the following hadith:
Narrated Umm Atiyyah al-Ansariyyah:
A woman used to perform circumcision in Medina. 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 Dawud, Book 41, Number 5251)
Various hadith define legal intercourse (for purity purposes) as occuring when the circumcized parts cross or touch each other. I.e. Circumcision of both men and women is presupposed.
Abu Musa reported: There cropped up a difference of opinion between a group of Muhajirs (Emigrants and a group of Ansar (Helpers) (and the point of dispute was) that the Ansar said: The bath (because of sexual intercourse) becomes obligatory only-when the semen spurts out or ejaculates. But the Muhajirs said: When a man has sexual intercourse (with the woman), a bath becomes obligatory (no matter whether or not there is seminal emission or ejaculation). Abu Musa said: Well, I satisfy you on this (issue). He (Abu Musa, the narrator) said: I got up (and went) to 'A'isha and sought her permission and it was granted, and I said to her: O Mother, or Mother of the Faithful, I want to ask you about a matter on which I feel shy. She said: Don't feel shy of asking me about a thing which you can ask your mother, who gave you birth, for I am too your mother. Upon this I said: What makes a bath obligatory for a person? She replied: You have come across one well informed! The Messenger of Allah (may peace be upon him) said: When anyone sits amidst four parts (of the woman) and the circumcised parts touch each other a bath becomes obligatory. (Sahih Muslim, Book 003, Number 0684)
The following traditions are found in Malik's Muwatta:
Yahya related to me from Malik from Ibn Shihab from Said ibn al-Musayyab that Umar ibn al-Khattab and Uthman ibn Affan and A'isha, the wife of the Prophet, may Allah bless him and grant him peace, used to say, "When the circumcised part touches the circumcised part, ghusl is obligatory." (Book 2, Number 2.19.73; see also No. 74 and 75) Yahya related to me from Malik from Nafi that Abdullah ibn Umar used to say, "When the circumcised part passes the circumcised part, ghusl is obligatory." (Book 2, Number 2.19.77)Yahya related to me from Malik from Yahya ibn Said that he heard Said ibn al-Musayyab asking a group of people, "What do you think about someone who has intercourse with his wife while he is in ihram?" and none of them answered him. Said said, "There is a man who has had intercourse with his wife while in ihram who has sent a message to Madina asking about it." Some of them said, "They should be kept apart until a future year," and Said ibn al-Musayyab said, "They should carry on and complete the hajj which they have spoiled, and then return home when they have finished. If another hajj comes upon them, they must do hajj and sacrifice an animal. They should go into ihram at the same place where they went into ihram for the hajj that they spoiled, and they should keep apart until they have finished their hajj." Malik said, "They should both sacrifice an animal." Malik said, about a man who had intercourse with his wife during hajj after he had come down from Arafa but before he had stoned the Jamra, "He must sacrifice an animal and do hajj again in another year. If, however, he had intercourse with his wife after he stoned the Jamra, he only has to do an umra and sacrifice an animal and he does not have to do another hajj." Malik said, "What spoils a hajj or an umra and makes sacrificing an animal and repeating the hajj necessary is the meeting of the two circumcised parts, even if there is no emission. It is also made necessary by an emission if it is the result of bodily contact. I do not think that a man who remembers something and has an emission owes anything, and if a man were to kiss his wife and no emission were to occur from that, he would only have to sacrifice an animal. A woman in ihram who has intercourse with her husband several times during hajj or umra out of obedience to him only has to do another hajj and sacrifice an animal. That is if her husband has intercourse with her while she is doing hajj. If he has intercourse with her while she is doing umra, she must repeat the umra she has spoiled and sacrifice an animal." (Book 20, Number 20.46.161)
The following Muslim website, Islam Questions & Answers, writes in response to a question regarding female circumcision:
Ibn Qudamah (may Allah have mercy on him) said, in his book al-Mughni: "Circumcision is obligatory for men, and it is an honour for women, but it is not obligatory for them. This is the opinion of many scholars. (Imam) Ahmad said: For men it is more strictly required, but for women it is less strictly required." (al-Mughni 1/70).
Circumcision of the female consists of the removal of a part of the clitoris, which is situated above the opening of the urethra. The Sunnah is not to remove all of it, but only a part. (al-Mawsuah al-Fiqhiyyah 19/28).
In this matter, it is wise to follow the interests of the female: if the clitoris is large, then part of it should be removed, otherwise it should be left alone. This size of the clitoris will vary from woman to woman, and there may be differences between those from hot climates and those from cold climates.
A hadith on the topic of female circumcision has been attributed to the Prophet (Peace and Blessings of Allah be upon Him), according to which he said: "Circumcision is a Sunnah for men, and an honour for women," but there is some debate as to the authenticity of this hadith. See Silsilah al-Ahadith al-Daifah by al-Albani, no. 1935.
How circumcision is to be performed is mentioned in the hadith narrated by Umm Atiyah, may Allah be pleased with her, according to which a woman used to perform circumcisions in Madinah. The Prophet (Peace and Blessings of Allah be upon Him) told her: "Do not abuse (i.e. do not go to extremes in circumcising); that is better for the woman and more liked by her husband." (Reported by Abu Dawud in al-Sunan, Kitab al-Adab; he said this hadith is daif).
The scholars' opinions cited above should be sufficient. (Source)
Further reading: