In this chapter, the people create an idol, Moses destroys it, and then the Levites kill fellow Israelites as punishment for their sin.
The first thing we see in this chapter is the "golden calf incident". We see in verse 1 that the people basically assume Moses is dead, since they last saw him over a month ago walk into this burning fire, and they figure he's not coming back, or at least, "we do not know what happened to him." Strangely, their response is to ask Aaron to "make us a god", and even stranger, Aaron goes along with this plan. Some revelry ensues, and the LORD sends Moses down to stop the madness.
According to the popular JEDP theory, the book of Exodus was written in various pieces between 900 and 600 BCE, with a principle purpose of eliminating idol worship and establishing the various principles we have seen (worshiping the LORD alone, avoiding idols, the Sabbath, etc). The struggle over the golden calf, then, is meant as a rebuke to the Israelite people of that later time, that they should give up idolatry.
Crucially, the Hebrews identify this idol with the LORD in particular (v. 5), suggesting that they did not actually intend to betray the LORD and worship another god, but rather felt that creating an idol was a proper way to worship their newfound sovereign.
Whether or not the JEDP theory is true, we can see that this chapter is certainly a rebuke of the Israelites in the story, because they were enamored of idols, regardless of which particular god they thought the idol was. Since the Israelites clearly worshiped idols at some point, I think given the context we can tell that they learned this pattern from the various peoples of the area, both Egypt and the Mideast at large. This again demonstrates the relative immaturity of the people in their worship of the LORD, whose ten commandments clearly delineate their behavior from those of the peoples around them. In other words, the ten commandments are an embodiment of the principle of separation, and idolatry is one of the things that they are supposed to be separated from. Here we see that they are not, in fact, separated from it, in spite of their earlier agreement to the covenant. I had previously remarked that the people appear rebellious towards Moses: now we are seeing another dimension to their immaturity, which the LORD treats as rebellion (v. 9-10).
One can reasonably infer from the text that the people and Aaron thought they were worshiping the LORD with sacrifices and an altar, that their behavior was appropriate and good, in particular when v. 5 says that they will celebrate a "feast to the LORD". The LORD's response seems pretty harsh in light of that, which is another reason why I think the author cares so much about the principle of separation. It is simply not good enough to worship the LORD in the manner of how other peoples worship other gods, for two big reasons. First, if the Israelites worship in the manner of other peoples, then there would be no distinction between the LORD and other gods, which would dishonor the LORD. Second, if there is no distinction between worshiping the LORD and other gods, then like an alcoholic taking just one drink, it would open the door to the Israelites returning to their idolatrous and polytheistic past. This would undermine the first commandment, that they would have no other gods besides the LORD.
In conclusion, we can see that at the heart of the principle of separation is the exclusivity of worshiping the LORD. The barriers designed to separate the Israelites from the other peoples is to isolate the Israelites from the spiritual adultery and polyamory of the peoples, who as we have briefly seen, are willing to worship any and every god that impresses them in some way. The LORD, as the sole and exclusive creator of the universe, desires primacy in the life of every person within the covenant. As I think I have made clear, this is completely contrary to the social and cultural standards of the time, which is most exemplified by the large and complex panthea of Greece and Egypt.
Idolatry is outlawed because in the Hebrews' immaturity in their new faith, they might be temped to relapse. Verse 4 says that the people exclaim the idol "is your god, O Israel", which establishes another reason why idolatry is outlawed: an idol is not meant to represent a god, an idol is considered (or at least spoken of as) the personage of that god, hence the expression, "make us a god".
This has two important effects. The first is that it constrains the god into the form of the idol, allowing the human makers to shape this god in whatever fashion they please, ultimately leaving the humans in power over the form of the god (while granting that god lordship over their lives in other areas). The second is that it limits the size and majesty of the god into the greatness of the form that the makers are able to construct. With reference to the LORD, he is reduced from the grandeur of lordship over all creation to a golden calf, a simple and inanimate creation.
After all the chapters of me pointing out the various manifestations of the LORD, whether human or clouds of fire and smoke, or thunder and lightning over a mountain, or burning bush, or whatever else we have seen, there are now people attempting to forge their own manifestation of the LORD, in their own wisdom and determination. It is at heart an act of extreme pride, that they would choose to know the LORD so well they can craft a form for him, just as they were once crafted in turn by the LORD, the maker of the universe. It should be evident to my readers after so much emphasis on the manifestations of the LORD that people trying to craft their own manifestation is a big problem, and it appears that the LORD thinks so as well.
Why a calf? Probably because that's what they would be familiar with, as a semi-nomadic herding people. Also, for whatever reason most of the idols in the OT are fashioned after animals, fish or birds, which we can infer from Ex 20:4, when it calls out the three realms of creation, which from Gen 1 we know are related to the three or four main populations, birds, fish, mammals and crawling things (snakes, insects, etc). This is probably the way idols were made during this era, that they would typically be shaped after various animals.
Lastly, since the Israelite people were explicitly commanded to not make idols in Ex 20, doing so here is a direct violation of the Mosaic Covenant, which is in turn an expansion and restatement of the Abrahamic Covenant. Since this covenant is what governs and defines the people's relationship with the LORD, violating it is a very serious offense even when ignoring all of the issues I've laid out above. Just like the people were splashed with blood when they agreed to it, the price of breaking the covenant is that one's own blood be shed in death.
Fortunately for the people, Moses intervenes on their behalf. I wonder what God would have done if Moses didn't intervene. Verse 10 states that God would destroy the tribes of Israel and build Moses into a great nation. Moses is a descendant of Abraham, so this would maintain the earlier promises. I also noted that the Mosaic Covenant was conditional on the Israelites' obedience, so performing idolatry could be reasonably construed as an annulment of the covenant. The main reason why I think that God wouldn't do this is the earlier prophecies of Jacob from Gen 49, which are predicated on the existence of the twelve tribes. However, these prophecies of Jacob are not directly affirmed by the LORD anywhere, so upon reflection I really do believe it's possible the LORD would have destroyed Israel.
Moses intercedes for the people, emphasizing that these are the LORD's people and that it would dishonor the LORD if other peoples saw the Hebrews destroyed. As a result, the LORD "changed his mind", which is not the first time this has happened (cf. Gen 6:6, the same Hebrew word "nacham"). While philosophers debate whether an omniscient being can change his mind, the bible clearly affirms that he can.
This passage is also frequently used to emphasize the humility of Moses, that he turn down the LORD's offer of greatness and instead intercede for the people who have given him so much trouble in the past (and future). I think this is true, but doesn't require much elaboration because the text is pretty clear and it's easy to understand what's happening.
Moses made the sons of Israel drink gold water possibly because of its bitter flavor, teaching them the "bitterness of the idolatry they committed", so to speak. This is just my speculation, as the underlying purpose is clear: Moses is angry, wants to destroy the idol, and force the people to identify with the wrong they have committed.
When Moses challenges Aaron, his immediate response is to shift blame to the people, which is not entirely unreasonable because by the account of verse 1, the people did pressure Aaron to make an idol for them. However, as the leader of the camp, Aaron was in a position to resist the people. Even if they killed him for it, it is his responsibility as the second in command and elder of the people who met with God in Ex 24 to refuse to commit idolatry and break the covenant. It is obvious from verse 2-5 that Aaron fully commits to idolatry, from planning and making the idol to declaring a feast celebrating the idol. Also, Aaron deflects blame by stating that the idol jumped out of the fire, rather than admitting that he forged it with a graving tool (v. 4). From Aaron's account, one could almost say he was an innocent victim, for who could have said that this magical golden calf would jump out of the fire when he threw some gold into it? And the evil people made him do it, anyway. This mostly reminds me of the various blame-shifting we saw in Gen 3 when Adam and Eve both deflected blame from themselves to another.
For whatever reason, Aaron is not punished for this indiscretion, but the people are punished as Moses commands the Levites to strike down their fellow Israelites. Moses declares that the Levites are "dedicated" to the LORD, and we will later see what this means (hint: it means they will serve the priests in the ministry of the tabernacle). Note that Moses and Aaron are from the tribe of Levi, so it is possible that they are banding with him out of their tribal bond to Moses rather than any particular zeal for the LORD.
Moses intercedes for the people again, asking for forgiveness, which the LORD seems to partially grant, not destroying the people entirely but "sm[iting]" them in v. 35. Yet, there is no more punishment at this time; rather, "in the day when I punish, I will punish them", a darkly ominous promise.
The Levites actions in v. 25-29 notwithstanding, I think we can see a broad picture of 1) the people quickly and unquestioningly return to their idolatrous roots the moment Moses departs, 2) the restraining influence of Moses both on the people, drawing them into the covenant, and on God, restraining God's wrath towards the people, 3) the obvious ineffectiveness of Aaron as Moses's delegated authority, 4) the LORD is obviously hostile to the people's rebellion, yet he allows himself to be restrained by the rather meager prayers of Moses. This whole episode is yet another instance where we see the prayers of a righteous man saving someone else, in this case, saving many someone elses. We saw the sacrifice of Noah in Gen 8:20-21 restrain the LORD's wrath that he would never bring a great flood upon the earth again, we saw the prayer of Abraham arouse the LORD's mercy for Lot and "righteous men" of Sodom in Gen 18:22-33, and now we see the prayer of Moses quelling the wrath of the LORD against his rebellious people.
While the most common message I have heard from this chapter is the humility of Moses in refusing personal glory, in my opinion it's just as important to note the impact of Moses's prayer to attain mercy for a people who don't seem to deserve it. This is especially interesting because Abraham's prayer was for the protection of righteous men, while Moses is praying for the forgiveness of wicked men, a wholly different affair, and it is a prayer that the LORD grants (while punishing the people later in the chapter). The claimed basis of this protection is the covenant with the patriarchs, and I already analyzed why the covenant could have been maintained through Moses, which leaves us with the honor of the LORD, that killing the people would dishonor him in front of other peoples.
However, I don't believe the LORD changed his mind because of logical persuasion, because God is omniscient and already knew everything that Moses was arguing (this is why philosophers question whether God can change his mind). Moses offers logical arguments, but we can deduce that logical arguments by definition cannot convince a being who is already familiar with those arguments. This leaves an alternative hypothesis, which is that God simply respects the opinions and prayers of his people, righteous people (both Noah and Abraham are called righteous) under the covenant, walking in accordance with his ways, and he will literally change what he was planning to do because of the words of these people.
This is derivative from the relational aspect of how God interacts with people, first established back in Gen 3 when it says that God walked in the garden of Eden, but reaffirmed in multiple places since then. Simply put, because God wishes to relate to the people made in his image, he will speak with them, listen to what they say, and upon request change certain actions or situations to fit their desires. That's not to say that God will do everything they ask, which I have already noted in this chapter regarding mercy and the punishment of the Israelites, but we also see that God genuinely listened to Moses and his actions reflect Moses's wishes. This is an immensely powerful principle and is the basis of prayer in Judeo-Christian religions.
If logical arguments do not convince God to do something, then why did Moses try them? Well, probably because Moses wasn't a philosopher and wasn't gifted with the plentiful analyses that we have today. :) In all seriousness, the reasons that Moses lists probably are important to God here, and another reason God wants to hear them from Moses before he changes his mind is so that Moses knows the reasons God is changing his mind. Remember, while we may think of Moses's prayer as Moses trying to change God's mind, God quite possibly views this prayer as a way for him to change Moses's mind. In general, that by us praying to God, we would ourselves realize the truth (or in some cases, the falsity) of what we are praying. Prayer is a two-way street, even if we don't realize it.