Notes (NET Translation)
9:1 On the eighth day Moses summoned Aaron and his sons and the elders of Israel, 2 and said to Aaron, “Take for yourself a bull calf for a sin offering and a ram for a burnt offering, both flawless, and present them before the Lord.
This chapter describes what happened after the seven days of ordination: the public cult is inaugruated by the newly invested priesthood.
The concentration of the entire chapter is upon the altar, as demonstrated by the curtailed description of the sacrificial procedure, which omits nearly every rite that is unrelated to the altar (e.g., the hand-leaning) but includes every rite involving the altar, even the most minute (e.g., the disposition of the suet pieces, vv 19-20), climaxed by the unique theophany upon the altar (vv 23-24). 
The elders are representatives of the people. Ironically, Aaron’s first sacrifice is a bull calf as if to atone for his making of the golden calf .
3 Then tell the Israelites: ‘Take a male goat for a sin offering and a calf and lamb, both a year old and flawless, for a burnt offering, 4 and an ox and a ram for peace offerings to sacrifice before the Lord, and a grain offering mixed with olive oil, for today the Lord is going to appear to you.'”
5 So they took what Moses had commanded to the front of the Meeting Tent and the whole congregation presented them and stood before the Lord.
6 Then Moses said, “This is what the Lord has commanded you to do so that the glory of the Lord may appear to you.”
The main purpose of these sacrifices is so that the Lord will appear. The glory of the Lord is his visible presence.
7 Moses then said to Aaron, “Approach the altar and make your sin offering and your burnt offering, and make atonement on behalf of yourself and on behalf of the people; and also make the people’s offering and make atonement on behalf of them just as the Lord has commanded.”
The role of high priest is now passed to Aaron. Aaron performs every kind of offering from Lev. 1-7 except those that were to be offered for specific offenses.
8 So Aaron approached the altar and slaughtered the sin offering calf which was for himself.
Although purification and burnt offerings had been offered for Aaron on the preceding seven days, he now had to offer one of each on his own behalf (vv. 8-14). His action in carrying out these sacrifices served as a public admission of his own sinfulness and need for forgiveness. The ritual for both sacrifices follows the normal procedure for burnt and purification offerings set out in chs. 1 and 4, except that the blood of the purification offering is smeared on the outer altar of burnt offering. Normally in the high priest’s purification offering the blood was smeared on the altar of incense (4:3ff.). The reason for this deviation may be that the incense altar did not yet need cleansing, since Aaron had not yet entered the tent of meeting where it stood. 
9 Then Aaron’s sons presented the blood to him and he dipped his finger in the blood and put it on the horns of the altar, and the rest of the blood he poured out at the base of the altar.
This is a detail of procedure missing elsewhere. In 8:14, 22 it is merely stated that the officiating priests “took” the blood. Practically speaking, it was necessary for another priest to assist the officiant; in Mishnah Yoma 4:3, 5:3, it is explained that a second priest held the sacrificial blood in a bowl and, later on in the ritual, handed it back to the officiant. 
10 The fat and the kidneys and the protruding lobe of the liver from the sin offering he offered up in smoke on the altar just as the Lord had commanded Moses, 11 but the flesh and the hide he completely burned up outside the camp.
12 He then slaughtered the burnt offering, and his sons handed the blood to him and he splashed it against the altar’s sides.
13 The burnt offering itself they handed to him by its parts, including the head, and he offered them up in smoke on the altar, 14 and he washed the entrails and the legs and offered them up in smoke on top of the burnt offering on the altar.
15 Then he presented the people’s offering. He took the sin offering male goat which was for the people, slaughtered it, and performed a decontamination rite with it like the first one.
16 He then presented the burnt offering, and did it according to the standard regulation.
17 Next he presented the grain offering, filled his hand with some of it, and offered it up in smoke on the altar in addition to the morning burnt offering.
18 Then he slaughtered the ox and the ram – the peace offering sacrifices which were for the people – and Aaron’s sons handed the blood to him and he splashed it against the altar’s sides.
19 As for the fat parts from the ox and from the ram (the fatty tail, the fat covering the entrails, the kidneys, and the protruding lobe of the liver), 20 they set those on the breasts and he offered the fat parts up in smoke on the altar.
21 Finally Aaron waved the breasts and the right thigh as a wave offering before the Lord just as Moses had commanded.
22 Then Aaron lifted up his hands toward the people and blessed them and descended from making the sin offering, the burnt offering, and the peace offering.
The lifting up of the hands was a posture of prayer . The blessing in vv. 22 and 23 may have been the blessing from Num. 6:24-27: “The Lord bless you and protect you; the Lord make his face to shine upon you, and be gracious to you; the Lord lift up his countenance upon you and give you peace.” Aaron descended from a ramp that gave access to the altar .
23 Moses and Aaron then entered into the Meeting Tent. When they came out, they blessed the people, and the glory of the Lord appeared to all the people.
Aaron may now enter the Meeting Tent.
It is unclear why Moses and Aaron entered the Tent at this time. The Sifra assumes that it was in order to pray for the anticipated appearance of God’s presence (kavod). Ibn Ezra states that it was in order to pray for the miraculous ignition of the altar fire. The two views are not inconsistent. When God’s presence appeared, a flame came forth from inside the Tent and ignited the altar fire, as we read in the following verse. 
Elsewhere, the glory of the Lord manifested itself as a pillar of cloud or fire .
24 Then fire went out from the presence of the Lord and consumed the burnt offering and the fat parts on the altar, and all the people saw it, so they shouted loudly and fell down with their faces to the ground.
The divine fire is proof of God’s acceptance of the new sanctuary and priesthood. While the sacrifices were already being burned  they would not have been fully consumed prior to the divine fire. The shout in question is a shout of joy . The ultimate goal of worship is to enter into God’s presence .
 Milgrom 1991, p. 571
 Ex. 32; Wenham 1979, loc. 1950
 Wenham 1979, loc. 1963-1968
 Levine 1989, p. 56
 Milgrom 1991, 586ff.
 Ex. 20:26
 Levine 1989, p. 57
 Ex. 24:16-17; 40:34-38
 Lev. 9:10, 13, 14, 17, 20
 Rooker and Cole 2000, loc. 4386
 Rev. 21:23
Hartley, John E. Leviticus. Dallas, Tex.: Word Books, 1992.
Levine, Baruch A. Leviticus. Philadelphia: Jewish Publication Society, 1989.
Milgrom, Jacob. Leviticus 1-16. New York: Doubleday, 1991.
Rooker, Mark, and Dennis R. Cole. Leviticus. Kindle Edition. Holman Reference, 2000.
Wenham, Gordon J. The Book of Leviticus. Kindle Edition. Wm. B. Eerdmans Publishing Company, 1979.