In this chapter, David places the ark in the tent and appoints Levites to praise God with music.
I think this chapter is straightforward, for the most part, and maybe even seems boring after a first reading, but I think there are a lot of small details that we can glean from.
For starters, we see David associating himself with the priests again. This is after 1 Chronicles 15:27 when David wore a linen ephod (the clothing worn by priests and Levites). Now in verse 2, we see David sacrificing the burnt offering and blessing the people.
The first part is very interesting because it's such a strong contrast to king Saul, who in 1 Samuel 13:9-10 offered a burnt offering and was harshly criticized by Samuel for doing so. Here king David is offering a burnt offering (the Hebrew words are identical in both passages), but he is regarded favorably for doing this. Why is it that Saul and David can both perform the exact same act, and yet the LORD is angry at one and pleased with the other? The text doesn't tell us directly, but I think the most likely explanation is that it has to do with the motivations and intent of the two people involved.
When Saul explains himself in 1 Samuel 13:11-12, he makes it clear that he was acting under great pressure because of his own men fleeing and the Philistines gathering for battle. Saul offers an ostensibly religious reason for offering the sacrifice when he says that he wanted to "ask the favor of the LORD", but from elsewhere in 1 Samuel we know that Saul was much more concerned with his appearance of religiosity before other men (for instance, see 1 Samuel 15:30-31). It's very likely the reason Saul offered the sacrifice was to rally the morale of his men and convince them that God would grant them success. He was using the LORD and the offering as a tool to further his own purposes, and the LORD rebukes him for it. Like the angel makes clear in Joshua 5:13-15, the LORD does not come and fight on our behalf (serving our objectives and purposes), he comes and challenges us if we are going to fight on his behalf, serving his objectives and purposes. He asks us to join his army, not to join our army. Of course, we know from elsewhere that the LORD blesses the people who follow him and even in the story of Joshua, we know that Israel was fighting in the promised land to take their inheritance. There are even places in the bible where it literally says that God is fighting on behalf of his people or on behalf of Israel.
I think the purpose of this story is to speak less about the intentions of the LORD and to speak more about the intentions of Joshua and Saul and David. We know that the LORD fights on behalf of his people: that's not the point. The point is, how do we perceive the LORD and how do we perceive our relationship to him? The contrast between Joshua and Saul could not be greater. Joshua bowed down before the angel of the LORD and asked, what do you want me to do? (Joshua 5:14). Saul, on the other hand, says that he disobeyed the LORD because he was afraid of the people (1 Samuel 15:24). Even though that is from a different story than when Saul offered the burnt offering, I think it speaks to his motivation and how he perceives the LORD. Saul is mostly concerned about maintaining his kingdom and keeping the favor of the people. He uses his faith as a means toward that end, and the LORD clearly objects to Saul's priorities. In 2 Samuel 6:20-22, David shows that he has little regard for what people think of him: he will leap and dance before the LORD even if it makes him look like a fool, because he was "celebrating before the LORD". Ironically, he claims that this will win him favor with the people, but it is not his intention or purpose. He lives before the LORD seeking to do what is right in the LORD's eyes, and that distinguishes him from people like Saul.
In conclusion, Saul offered a sacrifice to boost the morale of his troops; David offers a sacrifice to honor the LORD. Therefore the sacrifice of David is accepted and the sacrifice of Saul is condemned.
In the second part of verse 2, it says that David blessed the people. This is again David taking an aspect of priestly ministry, since priestly benediction was like... a thing. See Numbers 6:22-27 for an example. I think this is a similar story to what I wrote above. While David is perhaps infringing on priestly ministry, he is doing it from a heart of devotion to the LORD and therefore he is accepted.
In verse 3 when David distributions a portion of bread and meat to everyone in Israel, I think it's possible this was meant to be offered by the people as a peace offering, since Leviticus 7 explains that after a peace offering is made, the people were supposed to eat a meal, which was done in ancient times to confirm an agreement or covenant between two parties (for other examples, see Genesis 31:54, Exodus 24:11). Whenever the people offer a peace offering and eat the meal, they are affirming their covenant with the LORD.
Verse 4 explains the four different roles of the Levites: to minister, to celebrate, to thank God and to praise. In a single word, their role is to worship.
In verse 6, we see trumpets again. Trumpets play a variety of roles in Israel. In particular, I am thinking about the two silver trumpets in Numbers 10 that were used for organizing the camp as well as celebrating festivals. I am also reminded of the seven trumpets that were blown as the people marched around Jericho in Joshua 6:6. It looks like the trumpets are sometimes used for military purposes and they are sometimes used for celebration. Here, I think the trumpets are used for celebration as part of the worship.
There are a few minor I want to say about the psalm. This psalm shares passages with three different chapters from the book of Psalms, in particular Psalms 105, 96 and 106. It's almost impossible to say which was copied from which, whether this was the original psalm and those three psalms copied material from this one, or vice versa.
This psalm contains various references to the Pentateuch, such as the covenant between God and the patriarchs. Verses 19-22 seems to be a fairly clear reference to Genesis 20:6-7 when the LORD protected Abraham from Abimelech, which means that the author of this psalm must have been familiar with Genesis.
Verses 15-18 strongly affirms Israel's place in the land, which has an obvious significance to the returning exiles. Even the return from the exile is referenced in v. 35 when it speaks of gathering the people from the nations. As with much of Chronicles, this psalm is trying to explain why the promised land should belong to Israel and why God will bless them and establish them there.
Lastly, after the song concludes, David sends the priests and Levites to their stations. We see some of them left before the ark of the covenant in Jerusalem but others, including Zadok, are sent to the high place in Gibeon. Interestingly, Abiathar has disappeared from the record again; only Zadok is mentioned. It's also interesting that we see these two holy places. If my understanding is correct, I believe at this time that Gibeon is the home of the tabernacle of Moses, which is still treated as a place of worship, and Jerusalem is home to David's new tent with the ark. I think it's peculiar that they left the tabernacle in Gibeon rather than move it to Jerusalem, and I don't really have an explanation why. It may have been a political decision to help mend ties between Judah and Benjamin since Gibeon was located in Benjamin. At this point, Jerusalem is the political center of Israel, so if David centralized all the worship there also, Benjamin may have rebelled. That's my guess.