There are references in the Bible to a peace offering or fellowship offering. What was the purpose of this sacrifice?

The Peace Offering

The Peace Offering symbolizes fellowship with God. It was mainly offered for three different reasons. 

(1)  A votive offering or vow offering 

(2)  Thankfulness for a specific blessing such as thanksgiving for God’s deliverance in an hour of dire need.

(3) General thankfulness and fellowship (to be brought to one of three required religious services).

"Zebach sh'lamim" is the Hebrew term for this offering. Its root word is shalom, which means "peace" or "complete."

A VOTIVE OFFERING OR VOW OFFERING 

The term ex-voto meant “according to a vow” and originally referred to a gift given to fulfill a private vow to honor God. 

In the Old Testament, a votive offering was a voluntary offering vowed to God but not required by the Law. The technical term in Hebrew for such an offering is neder (vow). The only legislation prescribed for votive offerings regulated the place where they were to be made (Dt 12.5–6, 11).

The purpose of such a vowed offering was to give force to the prayer offered by the Israelite, and the formula in which the vow was couched contained either a positive or negative condition—negative, if the promise was to be fulfilled before the favor was granted, e.g., abstaining from wine for a certain period of time in order to gain God's blessing (1 Sm 14.24), and positive, if the Israelite promised to do something after Yahweh granted the favor. The positive vow always had as its object a cultic action. The Psalms contain frequent references to votive sacrifices that were publicly celebrated in the Temple at Jerusalem as a result of Yahweh's having granted a favor [Ps 21 (22).26; 49.(50).14; 55 (56). 13]. Nowhere is there any mention of good works or charity as the thing being vowed.

Hannah’s lavish offering when she dedicated Samuel to the Lord is an example of a peace offering given to commemorate the payment of a vow (1 Sam. 1:21–28). Psalm 22:22–31 was probably part of the liturgy that could be sung when peace offerings for deliverance or freewill peace offerings were given.

2 Samuel 24:25

David built there an altar to the Lord and offered burnt offerings and peace offerings. Thus the Lord was moved by prayer for the land, and the plague was held back from Israel.

Peace offerings were offered At the consecration of priests

Exodus 29:29

“The holy garments of Aaron shall be for his sons after him, that in them they may be anointed and ordained.

Peace offerings Were offered At the dedication of temple

1 Kings 8:62-64

At the coronation of kings

1 Samuel 11:15

Verse Concepts

So all the people went to Gilgal, and there they made Saul king before the Lord in Gilgal. There they also offered sacrifices of peace offerings before the Lord; and there Saul and all the men of Israel rejoiced greatly.

Source: https://bible.knowing-jesus.com/topics/Peace-Offerings

 For confirming the legal covenant

Offer of Thankfulness For A Specific Blessing

This is given to indicate sincere gratitude for blessings received and deliverance from difficulty. 

There was only a portion of the animal and cereal offered. What remained was given to the person presenting the offering as well as anyone within sight of the sanctuary (Lev. 7:11–18). 

Leviticus 7:15-16

15 “The meat of their fellowship offering of thanksgiving must be eaten on the day it is offered; they must leave none of it till morning.

16 “‘If, however, their offering is the result of a vow or is a freewill offering, the sacrifice shall be eaten on the day they offer it, but anything left over may be eaten on the next day.

During the eating of the sacrifice, strict cleanliness laws were observed (vv. 19-21). Despite being at peace with the worshiper, our Creator remained holy and could not tolerate impurity around Him.

General thankfulness and fellowship

Peace offerings Were offered At all the festivals

Numbers 10:10

Also in the day of your gladness and in your appointed feasts, and on the first days of your months, you shall blow the trumpets over your burnt offerings, and over the sacrifices of your peace offerings; and they shall be as a reminder of you before your God. I am the Lord your God.”

Source: https://bible.knowing-jesus.com/topics/Peace-Offerings

In coming to the Lord’s table, it is not merely to receive, but to offer praise, and thanksgiving, and worship, in holy fellowship. Thus, in speaking of fellowship or communion at the Lord’s table, the apostle refers to this very offering.

Thanksgiving Offerings Today

We do not need to sacrifice animals today as the first covenant along with its regulations have been fulfilled in Christ. However the offering of thanksgiving can still be made today without compulsion or obligation.  As His children, we have much to be thankful for.  In light of this, Paul wrote that we should be “giving thanks to the Father, who has qualified you for the inheritance of the saints in light” (Col 1:12) “because He has delivered us from the domain of darkness and transferred us to the kingdom of his beloved Son, because in him we are forgiven” (Col 1:13-14).

The goal of our lives should be to live lives that are holy and pleasing to God, which is why Paul wrote, “Let there be no filthiness nor foolish talk nor crude jokes, which are out of place, but instead let us give thanks” (Eph 5:4). Therefore I say to you and to me through Paul, "whatever you do, in word or deed, do it all in the name of the Lord Jesus, giving thanks to God the Father through him" (Col 3:17).  Does God have a plan for your life?  As God's will in Christ Jesus, "give thanks in every circumstance" (1 Thess 5:18).

Final Thoughts

God deserves our thanks.  He deserves our gratitude.  Christ has redeemed us "with His precious blood, like a lamb without blemish or spot" (1 Pet 1:19).  If we have peace with God and fellowship with Him through Christ's death, how can we not be grateful and freely offer and give thanks?  That is something to thank God for today and forever.