Method of slaughtering the animal
The animal should be watered and made to lie down on its flank so that its head is pointing towards the south and the muzzle in the direction of Makkah. Then one should take a sharp knife in the right hand and slit the throat just below the cartilaginous prominence (the “lump” in the throat) with the words “Bismillahi Wallahu Akbar.” This should be done with such force that the 2 carotid arteries, esophagus (food pipe) and the trachea (wind pipe) should be cut but the head should not be severed from the body.
Cutting these 4, viz. the esophagus, the trachea and the 2 jugulars is compulsory. As mentioned in Radd Ul Muhtaar:
(Radd Ul Muhtaar, Vol. 5, Pg. No. 207)
It is Mustahab (commendable) that the person on whose behalf the sacrifice is being performed should slaughter the animal himself/herself. If he/she cannot do so and someone else is doing so on their behalf, then it is better that he/she should be present at the time of slaughter. As mentioned in this Hadith:
It is narrated on the authority of Hadhrat Ali (May Allah be well pleased with him) that the Holy Prophet (Sallallahu alaihi wa sallam) told Hadhrat Fatima (May Allah be well pleased with her): O Fatima! Get up and stay with the animal that is being slaughtered on your behalf. Listen! With the first drop of blood that is shed, all the sins of the person on whose behalf the sacrifice is being performed are forgiven. Listen! In Qiyamah the sacrificial animal will be brought with 70 times its blood and meat. Then, it will be placed in the balance (Mizaan). Hadhrat Abu Sa‘eed Khudri (May Allah be well pleased with him) said: O Prophet of Allah (Sallallahu alaihi wa sallam)! Is this especially for the Ahle Bait (the family of the Holy Prophet (Sallallahu alaihi wa sallam))? They are worthy of every benefaction that is particularized for them or is this for all? The Holy Prophet (Sallallahu alaihi wa sallam) said: It is especially for the progeny of Muhammad (Sallallahu alaihi wa sallam) and generally for all.
(Kanz Ul Ummal, Hadith No. 12671)