As The Nelson Study Bible notes: "The Hebrew word for peace means 'wholeness, completeness, soundness, health.' When a person has this, in all its dimensions, that person is at peace. The peace offerings were times of feasting, drinking, talking, singing, and enjoying salvation as a great gift from God (see Leviticus 7:11-21 Leviticus 7:11-21  And this is the law of the sacrifice of peace offerings, which he shall offer to the LORD.
 If he offer it for a thanksgiving, then he shall offer with the sacrifice of thanksgiving unleavened cakes mingled with oil, and unleavened wafers anointed with oil, and cakes mingled with oil, of fine flour, fried.
 Besides the cakes, he shall offer for his offering leavened bread with the sacrifice of thanksgiving of his peace offerings.
 And of it he shall offer one out of the whole oblation for an heave offering to the LORD, and it shall be the priest's that sprinkles the blood of the peace offerings.
 And the flesh of the sacrifice of his peace offerings for thanksgiving shall be eaten the same day that it is offered; he shall not leave any of it until the morning.
 But if the sacrifice of his offering be a vow, or a voluntary offering, it shall be eaten the same day that he offers his sacrifice: and on the morrow also the remainder of it shall be eaten:
 But the remainder of the flesh of the sacrifice on the third day shall be burnt with fire.
 And if any of the flesh of the sacrifice of his peace offerings be eaten at all on the third day, it shall not be accepted, neither shall it be imputed to him that offers it: it shall be an abomination, and the soul that eats of it shall bear his iniquity.
 And the flesh that touches any unclean thing shall not be eaten; it shall be burnt with fire: and as for the flesh, all that be clean shall eat thereof.
 But the soul that eats of the flesh of the sacrifice of peace offerings, that pertain to the LORD, having his uncleanness on him, even that soul shall be cut off from his people.
 Moreover the soul that shall touch any unclean thing, as the uncleanness of man, or any unclean beast, or any abominable unclean thing, and eat of the flesh of the sacrifice of peace offerings, which pertain to the LORD, even that soul shall be cut off from his people.
American King James Version×). Paul describes Jesus Christ as our perfect peace offering (see Colossians 1:20 Colossians 1:20 And, having made peace through the blood of his cross, by him to reconcile all things to himself; by him, I say, whether they be things in earth, or things in heaven.
American King James Version×)" (note on Leviticus 3:1 Leviticus 3:1And if his oblation be a sacrifice of peace offering, if he offer it of the herd; whether it be a male or female, he shall offer it without blemish before the LORD.
American King James Version×). In the peace offering, unlike other offerings, the offerer was allowed to eat and thus find satisfaction in the offering: "God, man, and the priest, all fed together, all finding satisfaction in the offering. God first has His part and is satisfied, for He declares it to be very good. 'It is an offering made by fire of a sweet savour unto the Lord.' Man (in Christ) as offerer has his part, and is permitted to share this offering with his friends. And the priest, that is, Christ in His official character, is satisfied also, and His children are satisfied with Him. What a picture is here presented to us! The offerer feasts with God, with His priest, and with the priest's children" (Jukes, p. 108).
In this picture, not only is God satisfied but so is the whole priestly family—symbolic of Christ's family, the Church of God. So, too, is the offerer himself. Christ set the example here. Isaiah prophesied of Him, "He shall see the travail of his soul, and shall be satisfied" (Isaiah 53:11 Isaiah 53:11He shall see of the travail of his soul, and shall be satisfied: by his knowledge shall my righteous servant justify many; for he shall bear their iniquities.
American King James Version×, King James Version). Indeed, His life offering in God's service sustained Him as food. He said, "My food is to do the will of Him who sent Me, and to finish His work" (John 4:34 John 4:34Jesus said to them, My meat is to do the will of him that sent me, and to finish his work.
American King James Version×). Indeed, there normally had to be a "burnt offering" in order that a peace offering could follow. For the peace offering was to be offered "upon the burnt sacrifice" (Leviticus 3:5 Leviticus 3:5And Aaron's sons shall burn it on the altar on the burnt sacrifice, which is on the wood that is on the fire: it is an offering made by fire, of a sweet smell to the LORD.
American King James Version×). Being in alignment with God's will, as represented by the burnt offering, the worshiper would then be in a position to fellowship with God and with his or her family in the sharing of the peace offering.
As with the other offerings, there is much more to the peace offering that could be said—particularly when it comes to their being used in chapter 7 as thanksgiving or praise offerings or for taking a vow. Again, you are encouraged to study this subject on your own, as it is much broader than can possibly be covered here.