The Islamic method of slaughtering is that four or at least three of the veins in the throat of the animal be cut; the trachea (the wind pipe), the esophagus (a tube that directs the passage of food from the mouth to the stomach), the external jugular vein and the internal jugular vein. If three of the four are cut, then it is sufficient as the majority has the same rulling as the entirety. (Radd ul Muhtar’alad Durr ill Mukhtar). This is based on the statement of Allah’s Messenger (pbuh)

“Cut the jugular veins with whatever you wish.” (A similar narration is mentioned in Al Muwatta)

It is a condition that the slaughterer to be a Muslim or a Kitabi (people of the book). The slaughterer can be a woman, an unitelligent person or a sensible child who understands how to recite Tasmiyyah (mention Name of Allah (swt) and slaughter. Therefore, if an animal slaughtered by a polytheist who is not from the people of the book, by a fire worshipper, by an apostate, by a Jinn, by an insance person, or by an immature child then it is not Halal, even if any of them mentioned the Name of Allah swt and intented to gain proximity to Allah (swt) alone through their slaughtering. If a Muslim mentioned the Name of Allah (swt) and slaughters a goat for a fire worrshipper to devote it to his place of worship or he slaughters it for a non Muslim who will devote it to an idol, then it will be permissable to eat the meat of such an animal as the name of Allah (swt) was mentioned upon it, although it is disliked for a Muslim to do this. (Al Fatawa Al Hindiyyah)

Allah (swt) says,

“and what is the matter with you that you should not eat from that over which Allah’s name has been mentioned.” (Surah Al-An’am 6:119)

He (swt) also said in the Holy Qur’an,

“and the food of the People give the Book(s) is lawful for you.” (Surah Al Madiah, 5:5)