The first distinction of this kind of sacrifice is the fact that the gender of the sacrifice was irrelevant. Only males could serve as whole burnt offerings, but either males or females could be offered as peace offerings. The blood was still to be sprinkled on the brazen altar and a part of the animal was to be burned. Specifically, the fat, the kidneys and the connecting lobe off of the liver were to be burned.
God forbad the consumption of the fat and the blood of any beast (Leviticus 3:17). And, in the offering of peace offerings, He commanded that it be given to Him. Strangely, He called it His food (Leviticus 3:11 & 16). The rest of the peace offering remained for the priests and for the devotees to eat, but those particular parts (fat, blood, caul & kidneys) were retained by God.
But, you may ask, "Why the kidneys?" If you check the definition and application of the word kidney in this text you will find that in Moses' day it was used much like we use the word "heart." Not that they would necessarily have said, "I love you with all of my kidneys," but, the exact same word is translated "reins" almost as many times as it is translated "kidneys" (Psalm 16:7). So, if the people regarded the kidneys as the seat of emotion, affection or will... then it certainly makes sense that God would ask for it for Himself. The life of the flesh was in the blood, the excess sustenance of the body was energy stored in the fat and the prerogative of the soul was linked figuratively to the kidneys.