Circumcision for women is not obligatory
Question

I would like to know more about female circumcision in Islam. I have read the ahadith where the Prophet told a woman how to perform it. I would like to know if it is optional or obligatory, and - if it is obligatory - whether there is a certain way to do it (what part should be cut?).
Praise be to Allah. 
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-Mawsu‘ah 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-Da‘ifah 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 da‘if).
 
The scholars’ opinions cited above should be sufficient explanation.
And Allah knows best.