Leviticus 7:33
He among the sons of Aaron, that offereth the blood of the peace offerings, and the fat, shall have the right shoulder for his part. He among the sons of Aaron, that offereth the blood of the peace offerings, and the fat, shall have the right shoulder for his part.
The peace offerings were especially an expression of thanksgiving to God. The breast and the right shoulder of the peace offering belonged to the officiating priest as his allotted, specified share. The greater part was returned to the offerer to be eaten by him and his friends in a sacrificial feast. The fat alone belonged on the altar and had to be burned as an offering to God, not eaten, and the priests had to wait until this was done.
These strict directions guarded against confusion and were meant to prevent the priests from taking more than their rightful portion, so they had to be strictly heeded. The whole service, including the offering of the fat on the altar, pointed forward to the death of Christ. It was designed to preserve faith in the coming Redeemer and to direct all hearts to him in gratitude and faith, to the great Sacrifice that was to take away the sin of the world.
The sons of Eli broke this law by not being content with their allotted part. They demanded an additional portion and claimed whatever portion pleased them, refusing to wait until the fat had been burned. They threatened to take it by violence if denied, and used the great number of sacrifices at the annual feasts to enrich themselves at the expense of the people.