In this chapter, Jonathan tries to protect David and stands up against his father.
I think this is such an amazing chapter. There's a lot of stuff going on so I'll try to break it down piece by piece.
At the beginning of the chapter, David flees once more and returns to his friend Jonathan. In the previous chapter, Jonathan tried to reason with his father and openly opposed killing David. When Saul realized this, Saul started acting duplicitously, concealing his intentions from Jonathan. In return, Jonathan and David start acting covertly as well, with this elaborate scheme to have Jonathan "sound out" his father regarding David. The essence of this story is that if Saul doesn't care about David's absence, then it would show that Saul is not concerned about him. It's not like Saul really wants to have dinner with David, it's that Saul wants David to return to the kings residence so that he could be killed. If he is absent, then it's probably because David fled (which he does), and Saul is angry because he (rightly) perceives that Jonathan is helping to conceal David.
The particular circumstances of this chapter are uniquely Israelite, though. The new moon festival is one of the feasts prescribed in Num 28:11, which commands that Israel must make a special offering on the first day of every month. While it only says that an extra offering needed to be made in the tabernacle, in practice this became a day of religious observance and celebration, and as we can see in this chapter, it was customary for the king and his closest servants to gather for a meal on the new moon.
Another uniquely Israelite detail is that when Saul observes David's absence on the first day, his thought is that David is ceremonially unclean. As we read in the Pentateuch, there are many ways that someone can become ceremonially unclean. From being spat upon to touch a dead body or grave, there were many sources of ceremonial impurity. However, nearly all of the most common sources of ceremonial impurity resulted in only a single day of being impure, washing in the evening and thenceforth being clean. Some simple examples: touching the body of an unclean animal (Lev 11), having sex with a woman, an emission of semen, or touching articles contaminated by other kinds of bodily emissions (Lev 15), and the list variously goes on.
Saul's assumption is that David did something that made him unclean for a day, such as touching the dead body of some kind of animal, so he is not concerned when David is absent for the first day. But the feast goes on for a second day, and by then Saul knows that it is unlikely to a common impurity, though other kinds of ceremonial impurity last longer than one day.
Next, I think the story about Jonathan shooting the arrows and having a boy fetch them is interesting. You would think if Jonathan could just go outside by himself, he wouldn't need this elaborate ruse about "shooting at a target". For this reason, it is evident that Jonathan intends the arrow to be symbolic; he shoots an arrow "beyond" the boy to indicate that David must also go beyond, fleeing away like an arrow from the threatening presence of Saul. If the arrow is near, however, then David may also return like the boy returning with the arrows. I still think it's funny though whenever I read this chapter how Jonathan shoots an arrow past the boy, and when he reaches the arrow Jonathan insists that the arrow must be further away. The boy must have been a bit confused, seeing as how he was standing right near the arrow.
It's funny to me, but a tragic kind of funny because we can also see the raw emotion shared between Jonathan and David as David is being cast out like an arrow and they both weep over it, torn apart by the wrath and jealousy of Saul.
One of the more interesting things about this story is how Jonathan returns to his father. We have seen over the last 20 chapters how Jonathan is (in my opinion) so much more similar to David than Saul. They both have the same tenacity and bravery in battle, both have the same steadfast faith in the LORD. But something in Jonathan causes him to return to his father rather than join David in exile. This is more tragic than it might appear, because (spoiler alert) at the end of this book Jonathan will die alongside his father in battle, while David survives. If Jonathan had left his father he would have plausibly survived and would have remained a servant in David's kingdom. It is impossible to guess how David's reign might have been different with someone like Jonathan with him.
Even though Jonathan never joined in his father's evil plots, he would nevertheless destroyed by his father because for whatever reason, he simply found himself unable or unwilling to leave his evil family. It's like when Moses commanded the Israelites to separate themselves away from the tents of Dathan and Abiram, because these men were about to be consumed by God's wrath and anyone who was too close to them would become collateral damage (Num 16:26-27). It's not that God wants to kill innocents, but if you are too close to a big hole in the ground it just goes without saying that you are likely to fall into it. I think the story with Jonathan is similar; Saul was rushing towards destruction and even though Jonathan tried to do what he could to help David, he was never able to separate himself from the tents of Saul, and unfortunately Jonathan dies as a result.
As the story is developing, we can see Saul becoming increasingly violent such that he even tries to strike down his own son in his wrath. We can see the righteousness of Jonathan as he tries to protect David, but ultimately himself becomes a victim of Saul's destruction. David, for his part, is now finally and totally separated from Saul. He will talk with Saul a couple times, but will never return to serve in Saul's court or eat with him. This is the true beginning of David's exile, and Saul will spend the rest of this book hunting for David, a man who has literally never tried to harm him. And thus the cards are laid for how Saul's kingship will develop.
But I think my favorite part of this chapter is the raw emotions of it, because the emotions of the bible don't always show through clearly in the writing. But in this chapter I think it really does, as we can see David's desperation and his fear of impending death, telling Jonathan "there is hardly a step between me and death". Jonathan insists it is not true, and has an explosive confrontation with Saul. In the final part of this chapter, he meets David for a final, tearful goodbye. Just as David leaves Saul's house for the final time, this is also essentially the final meeting between David and Jonathan. We will read in 1 Samuel 23 that Jonathan visits David one last time, but at this time David and Jonathan were living in the same city, if not the same household, and from passages like verse 25 in this chapter, we can see that Jonathan and David would regularly eat meals together (with the king and Abner also present). So this chapter is not just the separation of David and Saul, it is also the separation of David and Jonathan. That's the tragedy inflicted by Saul's malice, these two young men who were so emotionally bonded being forced apart, the one cast out by a threat to his life and the other drawn in by familial obligation.