In this chapter, David organizes the military into 12 divisions and places officials over his royal property.
The previous chapter concluded David's organization of the Levites and priests, and this chapter basically organizes the military and his royal holdings. As with nearly every other chapter in 1 Chronicles, we should begin by asking "how does this relate to the temple?" I believe both of these systems (the military and the royal holdings) are listed because David is going to pass control over these systems to Solomon and he is commanding Solomon to use everything under his control to build and support the temple. From his royal holdings, David will directly give money to the temple project, and from the military Solomon will use them to maintain peace with the surrounding nations and possibly also to manage the conscripted labor that is used in the royal building projects.
Beyond the temple, we should also see this as part of David's handover of power to his son Solomon. All of these things are being listed and organized because this is what David is giving to Solomon as his official heir to the throne. In the next chapter we see David gather all these leaders together to swear them into obedience to Solomon, but this chapter is effectively a census of all the things that David (as king) will pass on to his successor.
With that said, I will now move on to more specific notes.
Most of the commanders in verses 2-15 are taken from David's mighty men (see 1 Chronicles 11), which makes sense given their prominence in the text. Verse 7 tells us that Asahel's son succeeded him, without mentioning why: Asahel died in the war against Ish-bosheth, Saul's heir, which David fought after Saul died. The war itself was omitted from Chronicles, but the Chronicler implicitly references it by naming Asahel's successor. Gad and Asher are left out of the list of tribes in verses 16-22. This is the only list of tribes I can think of where "Aaron" is given as one of the tribes. We also get the Ephraimites and two captains over Manasseh (the eastern branch and western branch). I can't really speculate what is the underlying logic for this particular selection of tribes, other than that it shows that Gad and Asher were probably non-factors in the time of Chronicles. It's possible these tribes had been completely wiped out by then.
Verses 23-24 refer to the census directly, and it's probably the same census as 2 Samuel 24 because it also discusses the "wrath" from the LORD that came upon Israel for doing this thing. Verse 23 is unique compared to 2 Samuel because it also explains the reason why wrath came upon them: counting the men was perceived as distrustful or disrespectful towards the LORD's promise to multiply Israel. By counting the men, David was perhaps saying that he wanted to see for himself if the LORD was growing Israel.
Verses 30-31 have an Ishmaelite and a Hagrite (i.e. Hagar-ite) responsible for David's flocks and herds. These are not descendants of Isaac, but more distant descendants of Abraham. We know from Genesis and elsewhere that the Ishmaelites are described as nomadic people and shepherds, so it makes sense that they are considered the most skillful shepherds and placed in charge of the kings flocks. What's funny is that when Jacob and his sons lived in Egypt, the Egyptians regarded Israel almost the exact same way, as these nomadic herders that are best suited for raising livestock (Gen 47:1-6), and Pharaoh actually asks that a "specially capable" man from Jacob's family should be responsible for the royal livestock, which is almost exactly the same thing happening here between Israel and Ishmael.
Verse 33 mentions Ahithophel, who sides with Absalom against David. Ahithophel commits suicide when Absalom is defeated, and evidently he is succeeded by Jehoiada and Abiathar. Again, Chronicles tells us who succeeds Ahithophel without telling us the grim circumstances of Ahithophel's death or his rebellion against David.
In verse 21, a son of Abner is made commander over Benjamin. We know from Samuel that Abner was a relative of King Saul and he fought as commander of the army for Ish-bosheth in the civil war against David. Abner killed Asahel and was himself assassinated by Joab. What this verse tells us is that David maintained a friendly relationship with Abner's family after his death, in spite of the possible bad blood that might exist between David and Saul's house. I think David has generally been very gracious to Saul's house, both because he refused to murder Saul twice when given the opportunity, and his friendship with Jonathan, and after Saul's death David has protected and provided for Mephibosheth. I would imagine that helped to mend ties at least somewhat.
Lastly, I want to talk a little bit about the military administration. Basically what we can learn from this chapter is that Israel maintains a standing army of 24,000 men who rotate through service on a monthly basis. Each man serves for one month a year and is otherwise part of the "reserve force". My guess is that the one month of service involves both training and generally patrolling the national borders and peacekeeping within the nation and possibly in the subjugated kingdoms around them. At the same time, it means that Israel can call to active service up to 288,000 men. Presumably that only happens in the case of war or national emergency.
Therefore, the vast majority of the men in Israel's army are non-professional soldiers. These are mostly farmers who are working on their land 11 months of the year and only serving in the army for one month. It's likely that the senior officers and commanders are professional but the majority of soldiers are not. From a cultural standpoint we can contrast this against later empires like e.g. the Romans who maintained large battalions of career soldiers. The difference is that if you have a lot of career soldiers, you need to plan and conduct an almost continual sequence of wars to keep them productive. That is, a permanent and professional military demands permanent conflict.
On the other hand, Israel has a relatively small standing force of non-professional soldiers and it only calls up a large armed body when needed. They still fight plenty of wars (as we have seen), and in the Mideast it seems almost impossible to avoid armed conflict even to this day, but I think it shows that economically Israel is not really based on armed conflict in the same way as later empires. Since most Israelites spend most of their lives farming and raising livestock, that is their nation's economic basis.
In the next chapter, David continues by charging his officials and military commanders to serve Solomon in his reign and in the construction of the temple.