Short answers, pulled from the story.
The Latin word sacrificium derives from sacrificus, which combined sacra (sacred things) and facere (to make or to do). It literally means to make something sacred. The term was later applied specifically to the Christian Eucharist, sometimes called a bloodless sacrifice to distinguish it from blood sacrifices.
Major theories include E.B. Tylor's gift theory, William Robertson Smith's communion theory from The Religion of the Semites, Emile Durkheim's dual social and divine communion theory in The Elementary Forms of the Religious Life, Sigmund Freud's psychoanalytic reading in Totem and Taboo, Rene Girard's scapegoat theory, Nancy Jay's argument that sacrifice maintains patrilineal structures, and Georges Bataille's conspicuous consumption theory from The Accursed Share. No scholarly consensus exists.
Current scholarly estimates place annual Aztec human sacrifice between a few thousand and twenty thousand people. Some sacrifices were performed to help the sun rise, others to bring rain, and others to mark expansions of the Templo Mayor in Tenochtitlan, the Aztec capital.
Muslims perform animal sacrifice annually at Eid ul-Adha in commemoration of the prophet Ibrahim's willingness to sacrifice his son. The animal, which may be a sheep, goat, camel, or cow, must be healthy and conscious. The meat is divided into three equal parts: one for the person performing the sacrifice, one for relatives, and one for the poor.
Maimonides argued in the Guide for the Perplexed that God always considered prayer and philosophical meditation superior to sacrifice. He treated God's permission of sacrifice as a concession to human psychology, given that the Israelites were accustomed to the animal sacrifices practiced by surrounding peoples. Nachmanides disagreed, arguing in his Torah commentary on Leviticus 1:9 that sacrifices were an ideal completely central to Judaism.
No. Roman Catholics, Eastern Orthodox, Lutherans, Methodists, and Anglicans all teach that the Eucharist is a sacrifice, though they differ on precisely what that means. Some Protestant denominations reject the sacrificial interpretation entirely, viewing the Eucharist as a holy meal rather than an offering.