In this chapter, the LORD establishes the annual day of atonement (Yom Kippur).
In verses 1-28 we are told the process for the day of atonement, and then in verses 29-34 we are told when these sacrifices should be made. Even though it's phrased a bit strangely, the "permanent statute" of v. 29 is simply a reference to what we were just told, the offerings of atonement for the people.
Yom Kippur is one of the most important ceremonies in the covenant, perhaps second only to the Passover. Both the Passover and Yom Kippur have similar purposes: the Passover is a remembrance of the people's deliverance from Egypt and the plague of death, while Yom Kippur is a more general atonement for sin. Both of them share the sense of "covering" with the Passover blood applied to one's doorposts, while the atonement blood is applied to the "kapporeth", the atonement piece on top of the ark of the covenant.
Probably the big difference is that the day of atonement is a collective confession and repentance of sin (for instance, v. 21 involves "confessing... all the iniquities of the sons of Israel and all their transgressions in regard to all their sins"), but the Passover is a bit more like a celebration of deliverance. Both of them are related to sacrifice and atonement, but they seem to me like two sides of the same coin: the Kippur side is confession of sin and the Passover side is celebration at the resulting deliverance and protection from God's judgment. They are positioned roughly 6 months apart on the Hebrew calendar, so that also possibly suggests an alternation between one and the other.
There are five different sacrifices listed for the day of atonement. This chapter is a bit confusing on the point, because v. 6 and v. 11 are describing the same sacrifice. The ram of the burnt offering in v. 3 is a different animal from the ram in v. 5. Basically, the priest brings a bull and a ram (v. 3) and then collects an entirely different ram and two goats in v. 5.
My understanding is that the high priest is to offer the bull first (like in v. 6), then cast lots over the two goats, and from there everything happens linearly after v. 12. This is a pretty convoluted description, and from the text it's not entirely clear that this is correct, but since there's only one bull being offered, it's the most logical construction. It is also supported by the Mishnah, an ancient Hebrew commentary on the Torah.
With that in mind, the five sacrifices are (in order): the bull of the sin offering, the goat of the sin offering, the so-called "scapegoat" (more on this later), and lastly the priest's burnt offering and the people's burnt offering (both rams, v. 24). The priest first atones for himself that he may atone for the people through the sin offerings, and the process concludes with the whole burnt offering of two rams (not to be eaten).
Of all of the many sacrifices we have read, I believe this is the first where some of the blood is taken behind the veil and offered upon the "kapporeth", literally the "atonement piece" (more discussion in Ex 25), and hence the name: its sole ritual purpose is atonement. We are told in the context of the death of Aaron's sons that "Aaron shall not enter at any time into the holy place inside the veil", i.e. that there are rules which must be followed, sacrifices which must be made, and he shall only enter for specific purposes. The death of Nadab and Abihu was a death of negligence or carelessness with respect to the divine ordinances, and as we have seen Leviticus is a book which intends to establish many such ordinances to govern the behavior and offerings of the priests. This chapter establishes the laws for entrance to the holy of holies behind the veil. We see first, there are many sacrifices (this should not surprise us any longer) and second, Aaron must fill the most holy place with a cloud of incense, that this cloud might "cover the mercy seat" to conceal the LORD's presence as the LORD dwells over the atonement piece on the ark, "otherwise he will die". I'm sure the threat of death is taken a bit more seriously now than before.
One other thing I thought of is that since only the high priest is allowed behind the veil, would that mean the blood on the ark of the covenant would never have been cleaned off? If so, it would just continue to accrete over the years, becoming a thicker and thicker layer, as it is sprinkled on every year. Eventually, the atonement piece and the ark of the covenant would have been completely coated in dried blood, and that layer would just keep growing thicker. It's a grisly thought, and yet virtually nobody would have ever been there to see it, since only one person in his entire generation would be authorized to go beyond the veil.
This is also the only ritual with a "scapegoat", literally "goat of removal" or possibly "the goat that departs", i.e. is sent away, or possibly "the goat of Azazel". The term "scapegoat" is a King James word that is a contraction of "escape goat", which conveys the same meaning as "goat of removal". The NASB again follows the KJV tradition and uses the term "scapegoat" from v. 8 and following. Unlike the other KJV terms I have highlighted (mercy seat, tabernacle), "scapegoat" actually conveys a pretty good sense of the Hebrew meaning to a modern reader. Scapegoat has diffused into modern English to mean a victim or sacrifice on whom blame is unfairly piled, and that's more or less what's happening here in Lev 16 as well.
An alternative translation is "goat of Azazel", considering the word "azazel" to be a proper noun rather than a word for "removal" or "departure". Some people consider Azazel to be a folkloric demonic figure, but I'm not going to discuss this in depth, so consider it another exercise for the reader to investigate as you wish.
The scapegoat in this chapter is first infused with all of the sins of the people by the laying of hands (v. 21) and is then sent away, for the purpose of bearing the sins of the Israelites out of the camp and away into the wilderness to die. The animal is not slain as part of the ritual, but tradition tells us that the Israelites wished to prevent the goat from wandering back into civilized areas and would therefore "send the goat away" over the edge of a cliff, ensuring its demise at the hands of gravity after being sufficiently pushed over the edge. The goat would have been ceremonially unclean, and this ritual goes to great lengths to ensure that its sole handler is sanitized by ritual washing before he returns to the camp, to prevent the impurity of the goat from leaking back in.
There have been many times I said that laying hands on the head of an animal sacrifice was a form of substitutionary atonement, and while I tried to justify it then, this chapter is really my prime basis for that assessment. Before (e.g. in the consecration of the priests, Lev 8:22, but many other places too), we were told that someone, the priest or the elders, laid their hands on the head of the animal, but we were not told why or what this meant. I argued it was transference of sins for two reasons: touching the animal is a type of association or identification and that we see a general pattern that sin or impurity can be transferred by physical contact. In this chapter, however, we are directly told that the sins and transgressions of the people are "laid on the head of the goat" (v. 21). Therefore I believe this is true for the earlier (and less specific) references as well.
In every earlier case, after transferring one's sins to an animal, the animal was killed as a sacrifice. In this chapter, the sins are transferred to a goat that is let live and driven from the camp, which is a bit surprising. Perhaps the idea is that since this animal bears the sins of the entire people from the whole year, they want to drive it away, as far as possible, so that these sins might not "pollute" the camp in a sense. I don't really know, but that's my theory.
Overall, I'd say the sacrifices in this chapter are generally pretty similar to what we have seen before, if a bit more somber and important. The few key differences are the scapegoat which is sent out into the wilderness, bearing the sins of the people, and the blood of the sin offerings which is taken into the holy of holies and sprinkled before the ark of the covenant. Other than that, the rest of the process is relatively ordinary and the burnt offerings and sin offerings are processed like they were described in Lev 1-5.