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We are permitted to eat kosher animals and fowl as long as they have been properly slaughtered. Kosher slaughter involves cutting the trachea and the esophagus as well as the adjacent arteries. Kosher slaughter is the most painless way to kill an animal. The carotid arteries, which supply most of the blood to the brain, pass through the very point where the slaughtering is done. Thus, immediately following slaughter, no more blood is supplied to the brain. This means by the time the pain signals would have reached the brain, the brain can no longer receive them. Even in the moments before that, the animal does not feel pain, because the slaughtering is done with a smooth, sharp knife and a quick movement. Fish, whose nervous systems are less developed, do not require slaughter.
Concern for animals stands out in other meat-related prohibitions as well. For example, eating part of a living animal is prohibited to both Jews and non-Jews. There is a mitzva to cover the blood of a slaughtered bird or wild animal. Since these generally live in the wild, covering their blood is a type of apology for our having trapped them. (The blood of a domesticated animal does not need to be covered after slaughter since it has been raised by people.) An animal and its child may not be slaughtered on the same day. Similarly, a bird may not be caught while it is sitting on its eggs (25:12 below).
