What does the peace offering in the Hebrew Bible represent?

Answer 1

The shelamim (Peace-offering) was an expression of thanks to God. Its Hebrew name is more closely related to "completeness" than to "peace," so its English name is a bit of a mistranslation. The Hebrew "completeness" means in the sense of wealth and satiety, for which it was offered up in gratitude. Various subcategories of this offering were made on the festivals, plus the day before Passover, plus whenever one had escaped great danger, plus voluntarily at any time.

More information:

One of the topics in the Torah (Leviticus ch.1-8) is that of sacrifices. Since the Holy Temple doesn't exist today, we don't now offer sacrifices. In earlier times, sacrifices were one of the ways of serving God; a type of manifest prayer. They consisted of bullocks, goats, sheep, rams, turtle-doves, or fine flour, accompanied by an offering of oil and of wine (see Numbers ch.15). Some offerings were obligatory and some were voluntary; some were for the individual and some were for the public. There was one offering which was intended to provide merit for the non-Jewish nations as well.
Some of the ideas included in the sacrifices are: gratitude (the Shelamim-offering, Leviticus ch.3), dedication to God (the Olah-offering, Leviticus ch.1), celebration (the Hagigah-offering during Festivals), atonement (the sin-offerings, Leviticus ch.4-5), and public worship (the offerings in Numbers ch.28-9).
Our traditions teach that as long as the sacrifices were offered in the Temple, the world was blessed (Talmud, Ketubot 10b) and atoned for (Sukkah 55b).
Some people have a misconception that the Prophets were not in favor of sacrifices. This is a mistake. The same Prophet Samuel who said that obedience is more important than offerings (1 Samuel ch.15), himself offered up sacrifices to God (1 Samuel ch.13).
What the Prophets meant was that repentance and obedience to God are essential; they should accompany the sacrifices and cannot be replaced by the sacrifices (or anything else). All of the offerings are intended to be part of a process of drawing near to God. This ideal of offerings together with repentance is explained in the Talmud as well (Berakhot 23a). The Talmud adds that, similar to the offerings, Torah-study and repentance also must go hand in hand (Berakhot 17a).
Another misconception is that Judaism changed after the Temple's destruction, and turned from a sacrifice-centered religion to a focus on prayer. It should be pointed out that Jews always worshiped in synagogues, even when the Holy Temple stood. Even within the Temple premises, there were several synagogues. During the Second Temple era, ancient Greek authors attest to the large synagogues that stood in all the countries where Jews lived. The Dioploston in Alexandria, for example, was famous for its size (Talmud, Tosefta Sukkah 4:6). In First Temple times also, synagogues served the same function that they do today (Talmud, Megillah 26b and Berakhot 31a).