the well-being offering: the shared sacrifice that celebrates a person's wholeness with God, in which God, the priest, and the worshipper each receive a portion and the worshipper feasts on the rest
This was ancient Israel's celebration sacrifice, almost always named in the plural (transliterated shelamim). What sets it apart is that it is the one sacrifice the worshipper actually got to eat. The animal was divided three ways: the fat and the blood went to God, certain choice cuts (the breast and the right thigh) went to the officiating priest, and the rest of the meat was eaten by the worshipper and their household in a glad meal at the sanctuary, in God's presence.
It was not a sacrifice for forgiveness. Unlike the offerings that dealt with sin and guilt, this one did nothing to repair a broken relationship; it assumed the relationship was sound and celebrated that fact. That is why it shows up at the high points of Israel's life: sealing the covenant at Sinai (Exodus 24:5), dedicating the ark (2 Samuel 6:17-18), crowning a king (1 Samuel 11:15), and the great temple dedication under Solomon (1 Kings 8:63). Wherever there was joy and thanksgiving before God, this offering turned it into a feast.
Leviticus 7 lists three occasions that call for one: thanksgiving after God rescues you, the paying of a vow, and a spontaneous freewill gift. These are three reasons for bringing the offering, not three different meanings of the word. In every case the sacrifice is the same and the heart of it is the same: a meal shared in God's presence.
The word's exact shade is genuinely debated (see the dispute below), which is why English Bibles render it so many different ways. The oldest Greek translation, the rabbis, and much modern scholarship point toward the idea of well-being or soundness: the worshipper, having received health, deliverance, or simple gladness from God, brings this offering to celebrate that wholeness at a table. The familiar renderings "peace offering" and "fellowship offering" each capture a real part of that, but each can also mislead.