Circumcision in Islam: Compulsory?
Question

Could you tell us what circumcision is and how and where it is done?
Praise be to Allah.Ibn al-Qayyim (may Allah have mercy on him) wrote a useful book on the rulings concerning the newborn, which he called Tuhfat al-Mawdud fi Ahkam al-Mawlud. In this book he wrote an extensive chapter in which he spoke of circumcision and the rulings thereon . The following is a summary of that, with additional comments from some other scholars. 
The meaning of circumcision in Islam (khitan)
Ibn al-Qayyim said: 
Khitan is a noun describing the action of the circumciser (khatin). It is also used to describe the site of the circumcision, as in the hadith, “When the two circumcised parts (al-khitanan) meet, ghusl becomes obligatory.” In the case of a female the word used is khafad. In the male it is also called i’dhar. The one who is uncircumcised is called aghlaf or aqlaf. (Tuhfat al-Mawlud, 1/152)
History of circumcision
Circumcision is the Sunnah of Prophet Ibrahim and the prophets after him. 
Al-Bukhari (6298) and Muslim (2370) narrated that Abu Hurayrah (may Allaah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Ibrahim (peace be upon him) circumcised himself when he was eighty years old, and he circumcised himself with an adze.”  
An adze (qadum) is a carpenter’s tool; it was also said that al-Qadum is a place in Syria.
Al-Hafiz ibn Hajar said: 
It is most likely that what is referred to in the hadith is the tool. Abu Ya’la narrated that ‘Ali ibn Rabah said: “Ibrahim was commanded to circumcise himself, so he circumcised himself with an adze and it was very painful for him. Then Allah revealed to him saying, “You rushed to do it before We told you what tool to use.” He said, “O Lord, I did not want to delay obeying Your command.” 
Ibn al-Qayyim said: 
“Circumcision was one of the things with which Allah tested Ibrahim, His Close Friend. He did them perfectly so Allah made him a leader of mankind. It was narrated that he was the first one who was circumcised, as mentioned above. What it says in al-Sahih is that Ibrahim circumcised himself when he was eighty years old. After him, circumcision continued among the messengers and their followers, even the Messiah. He was circumcised and the Christians affirm that, and do not deny that, as they also affirm that he was forbidden the flesh of pigs…” (Tuhfat al-Mawdud, p. 158-159)
Ruling on circumcision in Islam
But the scholars (may Allah have mercy on them) differed as to the ruling on circumcision. 
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said: 
“The most correct view is that it is obligatory in the case of men and Sunnah in the case of women. The difference between them is that in the case of men, it serves an interest which has to do with one of the conditions of prayer, namely purity (taharah), because if the foreskin remains, when the urine comes out of the urethra, some of it will collect there, and this causes burning and infection every time the person moves, and every time the foreskin is squeezed, some drops of urine come out, thus causing najasah (impurity). 
In the case of women, it serves a useful purpose which is to reduce desire. This is seeking perfection, not removing something harmful.” (Al-Sharh al-Mumti’, 1/133-134) 
This is the view of Imam Ahmad (may Allah have mercy on him). Ibn Qudamah said in al-Mughni (1/115): As for circumcision, it is obligatory for men and it is good in the case of women, but it is not obligatory for them. 
How is circumcision done?
Ibn al-Qayyim (may Allah have mercy on him) said: 
“Abu’l-Barakat said in his book al-Ghayah: In circumcision of a man, the skin at the tip of the penis (the foreskin) is removed; if he only removes most of it, that is permissible. It is mustahabb to circumcise females provided that is not done in an extreme manner. It was narrated that ‘Umar said to a woman who circumcised females, “Leave some of it if you circumcise (a girl).” Al-Khallal said in his Jami’: What is cut when circumcising: Muhammad ibn al-Husayn told me that al-Fadl ibn Ziyad told them: Ahmad was asked, How much should be cut in circumcision? He said, Until the glans (tip of the penis) becomes visible. 
Ibn al-Sabbagh said in al-Shamil: What is obligatory in the case of a man is to cut the skin on the tip of the penis until the entire glans becomes visible. In the case of a woman, it means cutting the skin that looks like the comb of a rooster at the top of the vagina, between the two labia; if it is cut the base of it should be left like a date pit.  
Al-Nawawi (may Allah have mercy on him) said: 
The well-known correct view is that everything covering the glans must be cut. (Al-Majmu’, 1/351) 
Al-Juwayni said:  
The hadith indicates that not too much of it should be removed (in the case of women), because he said, “Leave something sticking out and do not go to extremes in cutting.” (Tuhfat al-Mawdud, 190-192)
The point is that in the case of males, all the skin covering the tip of the penis should be cut, but in the case of females only a part of the skin that is like a rooster’s comb at the top of the vagina should be cut. 
Male circumcision: Why?
With regard to a man, he cannot be clean from urine unless he is circumcised, because drops of urine collect underneath the foreskin and he cannot be sure that they will not drip and make his clothes and body impure. Hence ‘Abd-Allah ibn ‘Abbas was very strict on the issue of circumcision . Imam Ahmad said: Ibn 'Abbas was very strict on this matter, and it was narrated that there is no Hajj and no prayer for him, i.e., if a person is not circumcised his Hajj and prayer are not valid. Al-Mughni, 1/115 
Female circumcision: Why?
With regard to the wisdom behind the circumcision of women , it is to regulate their desire so it will be moderate. 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) was asked about whether women should be circumcised or not. He replied: 
Yes, they should be circumcised, i.e., the top of the piece of skin that looks like a rooster’s comb should be cut. The Messenger of Allah said to the woman who did circumcisions: “Leave something sticking out and do not go to extremes in cutting. That makes her face look brighter and is more pleasing to her husband.” That is because the purpose of circumcising a man is to make him clean from the impurity that may collect beneath the foreskin. But the purpose of circumcising women is to regulate their desire, because if a woman is not circumcised her desire will be strong. Hence the words “O son of an uncircumcised woman” are used as an insult, because the uncircumcised woman has stronger desire. Hence immoral actions are more common among the women of the Tatars and the Franks, that are not found among the Muslim women. If the circumcision is too severe, the desire is weakened altogether, which is unpleasing for men; but if it is cut without going to extremes in that, the purpose will be achieved, which is moderating desire. And Allah knows best. 
Majmoo’ al-Fataawa, 21/114 
Paying money for circumcision: Permissible?
Ibn Qudamah said: 
“It is permissible to pay money for circumcision and for medical treatment. We do not know of any difference of opinion on this matter, because it is doing something that is needed and which is allowed in shari’ah. So it is permissible to pay money for it, like all other permissible actions.” (Al-Mughni, 5/314)
And Allah knows best.