In this chapter, the LORD instructs Moses on the ritual of the guilt offering.
Since this is the last of the five primary offerings, I will first discuss the guilt offering in particular and then conclude by summarizing the character and applicability of the five offerings in general.
The first thing we see in this chapter is a list of conditions where the guilt offering is required. This is similar to the sin offering, albeit more specific, because this chapter lists particular offenses that makes one "unclean".
The use of the word "unclean" is notable. Also, all of the things to make one "unclean" are referenced for the first time here. To be sure, they will be explained in more detail later, perhaps even more detail than some of my readers would want. But this is a legal document as well as religious, so the author had to consider many of the technical details so that the people could actually follow everything correctly. In light of this, and the general complexity of the law, perhaps the sin offering is starting to make more sense (i.e. an offering made by those who accidentally violate the law)?
Beyond that, this is the first time we read the word "unclean" in the OT. Previously we had seen the notion of "clean" with regards to Noah's sacrifices (Gen 7:2 and following), and now we are seeing the opposite of that, uncleanliness.
As I remarked at the time, "clean" doesn't mean free from dirt, although it possibly has that connotation. The OT largely uses these words, "clean" (Hebrew "tahor", pure, clean) and "unclean" (Hebrew "tame", foul, defiled, polluted) to refer to ceremonial purity. These rules can seem arbitrary and capricious when you read them without context, so I think the best example I can give is when the people washed themselves in preparation for the LORD's descent onto Mount Horeb in Ex 19. They weren't washing because God cares about their physical appearance. They were washing primarily as an outward sign of the inner purity and cleanliness that is required to meet with God. For one thing, being commanded to wash is supposed to change their mindset, to teach them that God is holy and that they have to prepare to meet God. That is, it is instructive for the people to wash, because it teaches them how to relate to God, how they must purify themselves to approach God. It is instructive to us as well, reading their example.
With that in mind, we understand that God was intending to dwell amongst the people in his tabernacle, his residence. As such, we can understand that there is an expectation or a requirement that the people must maintain a certain level of purity or cleanliness all the time that God dwelt amongst them. One must be even more clean or pure to approach the tabernacle itself, and that's one reason why the priests must wash before changing into the holy garments. In fact, the priests cannot wear their common clothes into the temple because the standard of holiness was too high. Cleanliness is simply a word to express that standard: to be clean is to follow it, to be unclean is to have broken it (accidentally or otherwise).
The purpose of the guilt offering then becomes clear. If a person somehow breaks the standard of cleanliness and becomes unclean then he can make a guilt offering to "clean" himself. It is important to note that being unclean is very different from violating the covenant. It is very possible to be unclean yet not violate the covenant and, depending on your interpretation, it is possible to be clean yet to break the covenant. This second point is hard to prove, but circumstantially, consider the golden calf incident of Ex 32. Nowhere in that entire chapter does God or Moses call the Israelites unclean for committing idolatry, yet the LORD was about to destroy the entire nation for what they had done.
We had been told a whole series of legal regulations governing the covenant in Ex 20-23, and neither "clean" nor "unclean" appeared anywhere in that covenant. Those were the laws which the people must follow to remain within the protective boundaries of the covenant. The violations here in Lev 5 (and later) that refer to cleanliness are rules that exist within the covenant. We can see that violating these rules have consequences in terms of offerings that must be made, but it categorically does not entail a violation of the covenant, which is punishable by death as a general rule . Instead, what we will find is that the laws of ceremonial cleanliness primarily dictate what status a person has to 1) enter the tabernacle, 2) dwell within the community. We had already seen precursors of the first in Ex 19 and when the priests were commanded to wash (Ex 29:4), so this should not surprise us. The second point (must be clean to dwell within the community) is related to the first, because as the LORD dwells amongst the people, his presence consecrates the entire camp of the nation. Just as one must be clean to approach the LORD, as a correlation to that one must be clean to be within the camp where the LORD dwells.
Since I'm now discussing the distinction between ceremonial law and covenantal law, I think now is as good a time as any to try to explain the various types of laws to my readers. I will start by listing what I believe are some of the various kinds of law in the OT and then some thoughts about them. In general terms, I think of about three kinds of law, though fully admitting that one could reasonably list more or less than that. The three kinds of law I consider are civil law, moral (or criminal) law and ceremonial law. In brief, civil law is the law that governs contracts, property disputes, and most anything that involves a dispute between two people that doesn't rise to a criminal offense. Some examples would be a contract between two people to sell wheat at a certain price, but one party decides to back out after agreeing to the deal. This is not covered by moral law, but the wronged party is still able to pursue compensation or demand an authority to enforce the contract (in modern terms, "equitable relief"). The Pentateuch is mostly not concerned with civil law. We know that Moses appointed judges who "judged all disputes" and maybe there are some provisions which are arguably civil, but that's roughly the end of it.
Moral, or criminal, law covers any action that breaks a moral law which is sometimes directed at another person (theft, murder) and is sometimes directed at God (idolatry, violating the Sabbath). In the modern world, criminal law is prosecuted by the state, but in ancient Israel, the state apparatus was extremely primitive, so criminal law was generally prosecuted by the community as a whole or by the family/clan of the wronged individual. Instead of a professional judiciary, the judges over these disputes (both criminal and civil) were the judges appointed by Moses, typically the elders of the people or the heads of the clans. Sometimes younger men were appointed leaders, as is the case with Joshua, but this would have probably been uncommon. Anyway, I could write a whole blog post on the Israeli judicial system, but for the sake of time and space I won't put it here.
The last type of law I will address is ceremonial law. Ceremonial law, as I said above, is primarily a set of rules that govern one's ceremonial cleanliness for the purposes of living in the community and approaching the tabernacle. It is never explicitly described as such by the bible, yet there are clear textual clues that support making the distinction between moral law and ceremonial law. For instance, the keyword "unclean" identifies many parts of Leviticus as being centered on ceremonial law. We can identify the ten commandments as being predominantly moral law, because they predate the tabernacle and any attendant ceremonies. From this basic framework, we can try to identify whether a given law is more like the ten commandments or more like the laws of cleanliness (which fill most of Leviticus).
It is very difficult to discern one type of law from another, because as I pointed out above, is the law against idolatry a violation of moral law or ceremonial law? Most people say moral law, because as I pointed out, it was contained in Exodus which is chiefly related to moral laws. Yet not every law is this clear. Case in point is Leviticus 19 which (without stealing my own thunder by analyzing it here) posits a peculiar blend of what modern scholars consider moral law and ceremonial law, commanding in v. 11 that "you shall not steal, nor deal falsely..." which are clearly moral commands as they were also within the ten commandments in Ex 20. But in the same chapter, v. 19 states that "you shall not sow your field with two kinds of seed, nor wear a garment upon you of two kinds of material", which is widely regarded as a ceremonial law.
Of course, the reason why it's hard for us to distinguish between moral and ceremonial law is that in most cases the author does not explain what kind of law he is talking about. In fact, even the words "ceremonial" and "moral" do not appear in the OT. That forces us to dig into context and innuendo when resolving what type of law is being given. Ironically, it is actually pretty important to figure out what is moral and what is ceremonial, because moral law is regarded as universal: it was wrong to murder before, and it's wrong to murder now. However, ceremonial law is situational: it is only necessary insomuch as one seeks to enter the tabernacle, which no longer exists. Even if the tabernacle did exist, the NT abolishes the Levitical sacrificial system so it would not be necessary to keep ceremonial cleanliness in any case (more on this when we get to the NT). This leaves us in what appears to be a rather awkward position: by assumption, we are required to follow moral laws from the OT but free to disregard ceremonial law. Without the OT clearly distinguishing between the two, we must dissect the OT ourselves by whatever criteria we can find.
This is a challenging problem, and (if I may be allowed a moment of sarcasm) it usually breaks down to: "the laws I want to follow are moral, and the laws I want to disregard are ceremonial", i.e. a post-hoc moral relativism, backfilling one's personal opinions into the text. Without a firm textual basis for dividing the laws, it is left as a matter of personal fiat. My hope is to do better than that, and my plan is to focus on context and intent. If we can capture the underlying purpose and motivation of the laws of the OT, this will enable us to more carefully analyze and apply them to the NT era in which we live. I will continue this discussion later when we are further into the bible and have more context for me to write about. Moving on.
Regarding the specific rules governing ceremonial cleanliness, that is the subject of large portions of Leviticus which are coming up, so this is not the last time we will discuss this subject. In fact, the rules listed in verses 1-4 are repeated later (at a much greater length), so I feel little need to explain them now. I will discuss these individually when I reach their corresponding sections. For now, I will simply note that while failing to testify leads to guilt, it is different from the commandment to not falsely testify (Ex 20:16). Even so, I will say I'm a bit surprised by the distinction because I would probably lump them together if I were writing a law.
Also note that all of these guilts are "hidden", so that the person did not know his violation at the time. If the person does know of their uncleanliness and addresses it more quickly, then (in general) a smaller sacrifice is required, and in most cases no sacrifice is required at all. The issue of hiddenness is important not because of intent (it is presumed that nobody will intend to become unclean), but because of the implied duration. In short, it is much more severe to be unclean for a long time than a short time, especially if you are unclean while still dwelling amongst the people in the camp. That's why a larger sacrifice is required.
Now I will discuss the details of the guilt offering itself. The guilt offering ritual is largely copied from the earlier offerings. Note that in verse 6, it specifies that one must sacrifice from the flock "as a sin offering", meaning using the ritual of the sin offering from chapter 4.
As with the burnt offering, there are several economic tiers for the guilt offering, scaling from a goat down to a small grain offering (by far the cheapest). Verse 7 tells us that the two doves or pigeons are offered as a sin offering and burnt offering, but since the sin offering does not have any provision for offering birds, it is explained to us how to do this in verses 8-9.
The grain offering is essentially identical to the grain offering of chapter 2, except without oil or incense "for it is a sin offering", and I guess this is meant as a sign of contrition on behalf of the offerer. Also this chapter tells us how much grain must be offered, which is different from the unspecified grain offering of chapter 2.
The chapter concludes with a command that any general violation of the "things which the LORD has commanded not to be done" must result in a guilt offering of a ram. Strangely we aren't told how the guilt offering must be performed, but it turns out that information is placed in Lev 7 for no reason that I can discern.
With that, I am done discussing this particular chapter and I'm going to summarize the five offerings. Since this post is already too long, I will be brief.
When you see burnt offerings, this is the "miscellaneous" offering that is used to cover a wide variety of scenarios. As such, it has the fewest definite connotations and is really just a way to give something to God.
Grain offerings are conceptually very similar to burnt offerings, and these two are frequently paired together. As with a burnt offering, both of these come from the "income" of the offerer, who would both grow his own crops and raise his own flocks. The Israelites obviously don't grow crops now (wandering the desert), but when established in the promised land, this will become a significant industry.
Peace (or fellowship) offerings are metaphorical "eating a meal with God" offerings. This is a sacrifice of communion.
Sin offerings are the offerings of personal or corporate atonement. While chapter 4 lists certain conditions when sin offerings should be made, this list is not exhaustive, and there are many other places in the OT where sin offerings are performed. The general rule is "atonement": a sin offering is a compensatory offering for some wrong committed, usually against the LORD (i.e. in the context of religious affairs). Note that many sin offerings are not made as atonement for a specific sin, but rather as atonement for general sinfulness. For example, the Passover was not atonement for a specific wrong, but rather was atonement for the Israelites' sinfulness in general.
Guilt offerings are similar to sin offerings, but typically more focused on ceremonial violations than moral or covenantal wrongs. Unlike the sin offerings, the guilt offerings are typically much more focused and uncommon in the biblical text. Of the ~40 references to "guilt offering" in the bible, more than half of them are in Leviticus, making this the least frequently encountered offering type in the bible. Of course, that doesn't mean they were infrequent in practice, just that it does not play a prominent role in the biblical text.
And that's it! With this framework laid, we are now ready to move on to the rest of Leviticus.
tl;dr: The OT has lots of sacrifices and laws and the death penalty and stuff like that.
 As a nomadic society, the Israelites could not materially support prisoners, because their society existed on razor-thin margins of survival. If you doubt me, consider all of the famines we have seen and how narrowly the Israelites escaped disaster time after time. Hence the focus on the death penalty: they simply could not afford to feed a set of people who did nothing to contribute to the survival of the nation as a whole. What's more, these people would have to be guarded, which costs even more time and effort. Compounded with the difficulty of relocating every few days (they are nomadic), and even keeping the prisoners from escaping becomes a serious problem. Critics love to excoriate the Israelites for their judicial system, but you have to keep it in context.