Leviticus 3, the Fellowship offering

The NIV study notes have a good explanation of this offering, and so I am quoting them here, after which will be some other comments from me.

From the Study Notes:
Two basic ideas are included in this offering: peace and fellowship. The traditional translation is "peace offering," a name that comes from the Hebrew word for the offering, which in turn is related to the Hebrew word 'shalom', meaning 'peace' or 'wholeness.' Thus, the offering perhaps symbolized peace between God and man as well as the inward peace that resulted. The fellowship offering was the only sacrifice of which the offerer might eat a part. Fellowship was involved because the offerer, on the basis of the sacrifice, had fellowship with God and ith the priest, who also ate part of the offering. This sacrifice -- along with others -- was offered by the thousands during the three annual festivals in Israel (see Exodus 23:14-17 and Numbers 29:39) because multitudes of people came to the temple to worship and share in a communal meal. During the monarchy, the animals offered by the people were usually supplemented by large numbers given by the king. At the dedication of the temple, Solomon offered 20,000 cattle and 120,000 sheep and goats as fellowship offerings over a period of 14 days (1 Kings 8:63-65).

The fellowship offering could be, unlike the others, male or female, and a sheep, goat, or one of the cattle. The animal had to be without defect, however, just as the other sacrifices were.

The reference to the 'fat tail' in verse 9 is actually a reference to a particular breed of sheep which is still widely raised in the Middle East. They do, indeed, have fat tails.

All the fat was the Lord's and to be burned on the altar as an offering to the Lord. This specifically included the kidneys with their fatty covering and the fatty layer over the liver.

This chapter closes with
This is a lasting ordinance for the generations to come, wherever you live: You must not eat any fat or any blood.