In this chapter, David and a descendant of Saul take parts of Israel and fight a war against each other.
A couple of chapters ago I said that not all of Saul's sons were killed in the battle on Mount Gilboa (in 1 Samuel 31), and now we can see that one of Saul's descendants, Ish-Bosheth, is declared king over the northern tribes by Abner. To understand what is happening, there are a couple things to keep in mind.
1) David is from the tribe of Judah. 1 Samuel 30 has David sending gifts to the elders of all the major towns in Judah. Judah as a whole is inclined to follow David because of the tribal affiliation to him, and this has been a persistent theme throughout the entire OT.
2) Abner is a powerful official under Saul, and it's pretty clear that he is supporting Ish-Bosheth because he wants to remain a powerful official. It's plausible that he is also acting out of loyalty to Saul, but there is no doubt in my mind that his own personal interests align closely with Ish-Bosheth, and that David becoming king would mean that Abner would be removed from power just as much as Ish-Bosheth. Furthermore, if Ish-Bosheth remains king, he would be indebted to Abner for supporting him, so Abner has everything to gain by "making him king".
3) Israel is likely supporting Abner and Ish-Bosheth because they represent the "current administration" and Abner is the commander over the nation's armies. At this point, David is just a well-known renegade commanding a couple hundred men, and he is supported by Judah because Judah wants one of their own people to be king. Abner is the commander over the entire nation's army, including tens of thousands of men. Simple momentum dictates that the nation would continue to follow Abner, even after the king is dead.
Even though Ish-Bosheth is anointed king over all the northern tribes, he is still clearly associated with Benjamin, which is his tribe and the tribe of his father Saul. This affiliation is so strong that verse 15 says that twelve men fought for "Benjamin and Ish-Bosheth the son of Saul".
What is unclear to me is why David and Abner are fighting a war. Without listing any grievances, it's as if both factions simply assumed that there could only be one king over a united Israel, and the natural thing for two kings to do is to try to kill or unseat one another. Another possibility is that Abner and Ish-Bosheth are possibly still trying to kill David, since they had spent at least a couple years chasing David in the wilderness. David had been an outlaw and renegade for a long time, and Abner had led the armies chasing after him. It might be that Abner is trying to kill David because he is afraid of David taking control of the nation, the same way that Saul was afraid of David.
I think all of these reasons are valid. Given how much respect David has for Saul, I think it's more likely that Abner would aggress against David than vice versa. So I guess this just goes to show, it only takes one side to start a war. But this is all speculation on my part. As my readers may observe, the text does not actually state a reason why they are fighting.
When they do fight, we see Abner and Joab sending forth 12 champions from each side, and all 24 men die at the hands of their foe. This is (in some ways) similar to Goliath and David, who fought as champions on behalf of their respective side. Just like when David and Goliath fought, these men fight each other as the prelude to a battle, not as the replacement for a larger engagement.
The rest of this battle has some interesting and important details. Before discussing that, I will remind everyone who all these people are.
Joab, Abishai and Asahel are sons of Zeruiah, and they are allies of David. Joab in particular is basically David's chief commander, equivalent in rank and purpose to Abner. Abner, for his part, was the chief commander under Saul, and is now the chief commander under Ish-Bosheth. Since Ish-Bosheth was not the heir apparent (that was probably Jonathan), he would not have been groomed for leadership. In fact, this is the very first chapter where Ish-Bosheth is even mentioned. So he was clearly not a senior or visible figure under Saul, and while Abner "made him king", in many ways Abner is a more powerful figure now than Ish-Bosheth is.
Abner kills Asahel, but he knows that Joab is unlikely to forgive him for this. Indeed, I'm going to be talking about this later because there will be conflict between Abner and Joab in the next chapter.
In the end, Abner and his men flee to Mahanaim (which is where Ish-Bosheth reigns, see v. 8). Joab and David's men return to Hebron where David reigns over Judah.