In this chapter, God gives Moses the ten commandments and other laws.
Verse 2 indicates that these laws are to be followed because of how the LORD brought freedom to the Israelites. This assertion is repeated over and over in the Pentateuch, which is perhaps fitting because this is the book supposedly written by Moses after their departure from Egypt.
The ten commandments are certainly the most famous laws in the Pentateuch, but as we have seen they are not the only laws. The ten commandments are important however as they are the laws later written on the stone tablets, quoted by Jesus, and embody the central principles of the Hebrew faith. The first five commandments are the religious commands, related to their faith, and the last five commandments are the moral laws, governing Hebrew society. Obviously the Hebrews had no concept of separation of church and state as so many Americans desire these days.
Almost all of these commands are new to the bible. The only command that appears to be repeated is to honor the Sabbath. However, note that the Sabbath here is much more expansive than the one in Ex 16. In chapter 16, the only command is that the people would not gather manna: here, the people are commanded to not do any kind of work.
The first commandment is exclusive devotion to the LORD. The ancient Mideast was polytheistic, so it would have been common for a single person to worship multiple gods. We see this sort of flexibility in how other peoples relate to the LORD, such as Melchizedek (Gen 14) and Jethro (Ex 18). While they are not devoted to the LORD, they are happy to offer him a few sacrifices to honor his military victories. For its part, Egyptian society has a whole cast of gods, just as the later Greek nation-states built an elaborate hierarchy of deities. In Canaan, many races had specific patron gods, but they also intermixed with other gods. The command that the Israelites would worship their LORD alone is therefore part of the principle of separation, keeping them away from the malign influence of their neighbors' faiths.
The second command is outlawing the creation of idols, graven statues to represent their god(s). This is another common Mideast religious practice. Since the first command is against having other gods, this verse most specifically prohibits creating an idol of the LORD. While no specific reason is given for this command, in my opinion it gets back to the topic of the various manifestations of the LORD. I think the LORD wishes to control how he appears to people so that he can build and influence their opinions of him in this way. If people create idols of him, then they can shape those idols and develop their own image of who God is. This is an error of presumption, because only God can tell us who he is.
Verse 5-6 are very interesting and much has been written about this passage, especially the notion punishing children for the iniquity of their fathers. Later in the Pentateuch we will see that God specifically prohibits punishing children for the sins of their fathers. However, we also earlier read the LORD command Israel to never forgive the Amalekites and to pursue them to destruction because they attacked Israel. Many people dislike this because it goes against our modern notion of personal responsibility, that children are not responsible for the actions of their fathers. As I said, the OT partially supports this position as well. I think there are two reasonable answers I can give to this critique. First, parents guide and shape their children in many ways obvious and subtle. So even if children are not directly responsible for the sins of their fathers, it is frequently the case that they walk in the same sins and therefore deserve the same punishment. Second, the bible does not have the same individualistic notions as modern western society. Consider the example of Abraham, how he passed down the divine promise to his son, from Isaac to Jacob, from Jacob to his children, and all the way down to Moses and the sons of Israel. Isaac did nothing of his own to receive the promise: he was given it by God in honor of his father Abraham.
To many westerners this seems unfair, but in some respects it's no less unfair than the fact that different people have parents who guide them in very different directions. I myself had a family who guided me to get a college education, and in this way I was blessed by my family through no action of my own. If this is unfair, then life is unfair, but that doesn't have to do with the bible. I think the more important question here is where God is fair in his judgments of people's lives. I believe the answer is yes, because God judges people both by what they do and also by what they have been given. You cannot look at one without looking at the other. End tangent.
The third command is just another form of reverence, to hold the name of the LORD in high regard and to use it carefully. Historically, the Jews treated this command so severely that the proper pronunciation of of YHWH has been lost, because they refused to speak it under nearly any circumstances. This is also why many Jews today write God as "G-d"; they omit the middle character so that they do not even say the word God, in fear of violating the 3rd commandment. I think it is unduly inflexible to not use the word God to describe God, and this is a small example of the big differences between the Law of Moses as it was written and how Christianity is practiced today. I personally still adhere to the 3rd commandment, but that does not preclude speaking the name of the LORD in any situation, only when such speaking is disrespectful or dishonorable.
The fourth command is a restatement of the Sabbath, which as I said above is an expansion of the command in Exodus 16. This passage specifically ties it to Genesis 1 and the creation of the universe.
The fifth command is to honor one's parents. It's peculiar that this is grouped together with the religious commands rather than the moral commands, but that is how the commands have traditionally been divided. I don't really have anything else to say about this command.
The next five commands cover various aspects of moral fault: murder, adultery, theft, false testimony (in the context of legal proceeding), and covetousness (or, desire for the possessions of another). The first four have been pillars of western morality for a long, long time. Only recently has adultery exited the purview of judicial action, though it is still considered a fault for the purposes of divorce (i.e. gives a legal advantage to the wronged party). The other three, murder, theft and false testimony while under oath, are still illegal in most countries.
I think the last command, covetousness, is more interesting because the first four moral commands outlaw various actions. The last one outlaws various desires, which is inherently impossible to enforce. It's similar to the first command which is also more of a mindset (desiring other gods) compared to the various other religious commands.
Keep in mind, there is no stated enforcement mechanism for these laws. While it's true these laws become the governing constitution of Israel, there are no stated punishments here because it is a contract that they are making with God himself, and so God is the ultimate authority and judge of the Israelites' behavior. It stands to reason that God can understand the desires of the human mind just as much as he knows the intricacies of the human body, for he has created all of these things. While there are no specific punishments stated, we know that it is by following these commands that the Israelite people would be the LORD's special possession, a royal priesthood and holy nation. The ultimate punishment for violating these laws is the broken covenant and the loss of the covenantal blessings. Note once again that this covenant is corporate and treats the Israelite people as a singular thing. This is why those people who do not keep the Passover must be cut off from Israel, because if they remain within the nation then the nation as a whole becomes liable for the broken covenant.
After all this, the people reiterate their fear of the LORD and ask Moses to intermediate between them and the LORD, as he has previously done (v. 19). I have heard some commentators say that by asking this, the people have abdicated their "royal priesthood" from Ex 19:6, but I don't believe this is accurate. Moses was already speaking to the people on behalf of the LORD for the entirety of Exodus up to this point, and the royal priesthood of 19:6 was conditional on their fulfillment of the commandments, so it was not yet a present reality. In the future, if the people hold to the commands then they will receive the promised blessing.
I don't know what v. 25 refers to when it says that using tools on a stone altar profanes it. I have never heard a good explanation for this peculiar belief. Verse 26 is probably a reference to people seeing the "nakedness" under a priest's garment if he is climbing up steps, since they possibly didn't wear pants or underwear.