Wa `alaykum as-salam wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother in Islam, thank you for your question, which reflects your care to have a clear view of the teachings of Islam. Allah commands Muslims to refer to people of knowledge to get themselves well-acquainted with the teachings of Islam as well as all aspects of life.
First of all, the rules of Islamic slaughter are quite simple and precise. They can be summed up as follows: invoking the name of Allah, cutting the throat or arteries with a sharp knife, thus administering a quick death and sparing the animal unnecessary torture and suffering. This is to be done in such a way that the blood is drained.
These are the rules of slaughter as clearly outlined in the pristine sources of Islam, namely, the Qur'an and the Sunnah; these are essentially the same rules that have been formulated by the great imams and jurists in all the authentic schools of jurisprudence.
In his response to your question, Sheikh Muhammad Nur Abdullah, former president of the ISNA (Islamic Society of North America) and member of the Fiqh Council of North America, states,
To briefly explain it for you, the rules of slaughtering have been driven from the sayings of Prophet Muhammad (peace and blessings be upon him). Based on the interpretation of these sayings, the scholars have agreed on some conditions of slaughtering, while differing on others. All scholars, for example, agree that the animal must be slaughtered, i.e. the animal is alive and it does not die before the actual slaughter.
Jurists, however, differ on stipulating that the name of Allah be mentioned at the time of slaughtering. Some scholars, basing their argument on verse 121 of Surat Al-An`am, forbid the animal that has been slaughtered without mentioning the name of Allah. Other scholars, basing their argument on verse 5 of Surat Al-Ma'idah, allow eating the animal that has been slaughtered by the People of the Book, i.e. Jews and Christians, and restrict the verse in Surat Al-An`am on polytheists and idol worshippers.
Stunning in most cases does not kill the animal but rather numbs the nerves so that the pain is lessened. Therefore, it is allowed to eat the animal slaughtered by the People of the Book provided that you mention the name of Allah before eating.