This is part of the dictionary article I'm writing:
Theories abound to explain how sacrifice functions in religion. A key element in most of these is sacrifice as a mediating factor in the relationship between gods and mortals. Certainly sacrifice functions this way in the Old Testament. Yet within the pages of the Old Testament we find more than one sacrificial paradigm, and thus more than one perspective on sacrifice as an instrument of mediation between God and humankind.
On one end of the spectrum, embedded within the biblical narratives, are glimpses of Israel’s pre-scriptural past. We hear rumblings of child sacrifice (e.g., Ezek. 16:20-21; 1 Kings 11:7; cf. Deut. 13:2) and puzzle over God’s command that Abraham offer up Isaac (Gen. 22:1-19). We are unsure what is taking place when Zipporah touches Moses with the foreskin of her son so that God does not kill her “bridegroom of blood” (Exod. 4:24-26). Similarly unfamiliar is the strange ritual where Abraham cuts a heifer and a goat in two and lays with them an uncut turtledove and pigeon (Gen. 15:12-21). Then when he is sleeping, the LORD reveals the future and a flaming torch passes between the cut pieces. These obscure artifacts of Israel’s pre-biblical past play into the hands of sacrificial theories that connect sacrifice to human violence and the appropriation of divine power on the human plane (cf. 2 Kings 3:26-27). Sacrificial mediation in such instances has less to do with the person of God than with the intrinsic power of the sacrificial acts themselves.
In terms of mediating value, the two most important functions sacrifices can perform are 1) their function as gifts to God and 2) their function in expiating sin and uncleanness. Since all sacrifices and offerings involve the presentation of something to God, the gift aspect of sacrifice is perhaps its most fundamental characteristic. It is no coincidence that the most generic Hebrew words for sacrifice have the connotations of a gift: minchah, mattanah, and qorban.
As gifts, sacrifices perform the same meditating functions that gifts do between humans. Some are offered to secure God's favor, such as when Saul offers a sacrifice before going to battle (1 Sam. 13:9). Votive sacrifices and offerings are either offered when making a vow or upon its successful completion. Jephthah's sacrifice of his daughter is such a sacrifice after he returns from a victorious battle (Judges 11:30-31, 39). The gift of a sacrifice can propitiate God's wrath toward an individual or a people. After the LORD sent a pestilence on Israel for David’s census, God’s anger is finally satisfied when David erects an altar on the threshing floor of Araunah the Jebusite and offers burnt offerings there (2 Sam. 24:15-25).
In this same vein sacrifices are offered regularly to maintain God's favor and to give him thanks for good things. Such gifts function somewhat like the tributes one might give to a king for his protection and blessing. The gift of the firstfruits of harvest give God thanks for the way he has blessed the soil and look to future blessing. Sacrificial gifts thus mediate good relationships between God and humanity in ways that humans can easily understand.
The function of sacrifices in relation to sin and impurity is both one of the most familiar and yet most misunderstood. In general, sacrificial law was not designed to restore an individual's relationship with God for intentional sin, so called "sins with a high hand" (e.g., Num. 15:30-31). The chatta't or "sin offering" thus more mediated impurity and uncleanness than what we would consider "moral" sin, the distinction between the two being very loose in Old Testament times (cf. Lev. 4:2). A person "impaired" by things like childbirth, leprosy, or the inadvertent violation of a taboo might be restored to wholeness by way of this "purification offering."
The idea that the sacrificial animal substituted for the person sacrificing perhaps has some element of truth, but is prone to overstatement given modern debates about penal substitution. In the Day of Atonement ritual, we should probably think more of defilement being transferred from the community of Israel to the animal, which then takes that miasma out of the community and into the desert.
Modern notions of guilt seem anachronistic in relation to the Levitical system of Israel. The so called "guilt offering," the 'asham, functioned almost like a fine, a penalty offered because one has inadvertantly wronged God (cf. Lev. 5:14, 17). Yet even in this case, the amount of the sacrifice is not gradated in accordance with the level of the offense. It thus does not figure well as some kind of substitution for the offender.
In general, the mediation provided by these kinds of offerings had far more to do with cleansing impurity, as the verb kipper seems to indication, perhaps having a fundamental sense of wiping away...