In a tabletop roleplaying game, the PCs are the main characters. This is an important conceit. Players should feel like they’re important. But also they’re usually boring, static character who very stubbornly resist anything like a character arc. Players usually do not create characters with the intention that they will change and evolve in any way except through ever-growing power. If your group is all really invested in narrative and roleplaying and such, you might all sit down together and intentionally create a party where this is not so. Probably your group doesn’t want to do this, though, which means the narrative has be carried entirely by NPC character arcs, and it has to do that without taking the spotlight off of the PCs.
This is tricky, because the PCs are boring, but fortunately you have an ace up your sleeve: Anything that happens to a PC is automatically going to be more interesting to the player running them, because it is happening to them, specifically. While ordinarily having the role of pushing the plot forward handed off to a character who is not changed by that plot at all would be terrible, for the specific medium of TTRPGs it’s actually totally fine.
Side note: There are exceptions to this rule even amongst books and movies. Batman, Paddington Bear, and Samurai Jack change the world around them rather than being changed by it, putting other characters through arcs by forcing their flaws into conflict with their redeeming qualities and hoping their noble ideals triumph over their base instincts. If you want to pull this off outside of a TTRPG, your static protagonist needs to stand for something and be so unwaveringly committed to that stance that they can bend the world around them. As mentioned earlier, though, for a TTRPG your audience will cut the protagonists insane amounts of slack just because they designed the protagonists themselves.
Other than handing off responsibility for plot momentum to the PCs, an NPC’s arc works mostly the same as a (regular, non-static) movie or book protagonist’s would. I’ll drop the standard disclaimer here that this should not be treated as a formula to be rigidly adhered to but rather as loose guidelines that should be followed when they work and discarded when they don’t, although if you need a blog post to tell you that then you’re probably at the level of craft where following guidelines exactly is your best bet anyway.
In order to establish how an arc changes an NPC, we first need to establish what they’re like in advance. Establishing normalcy doesn’t have to last longer than a scene if the arc is intended to last for only one adventure. In fact, if you want to run an NPC through an arc in the space of a single scene, you can (although it’s hard), in which case establishing normalcy could be pulled off with as little as one sentence. What’s important is that the NPC communicates to the players who they are, how they do things, and why they either do or do not wish to change their circumstances. The NPC needs to establish two separate motivations that right now aren’t in conflict with one another and can coexist. A cop wants to uphold the law but also loves his family. Easy to see how those two motivations can coexist without coming into conflict, and it’s easy to imagine how both of those could come up in a scene with the PCs. He’s a quest ATM telling them what information he needs on the evil ninja-themed gangsters wreaking havoc on the streets, and he’s in a rush about it because he’s got a family function of some sort to get to and he really doesn’t want to miss it. This is the kind of flavor that helps spice up an interaction even if you aren’t setting up a character arc at all.
After this, circumstances change. As with each of these steps, there’s a million tiny variations on how to do this (and while the distance between similar variations is tiny, the distance between either end of the spectrum is huge). Maybe the story here is that the NPC got the change they wanted and it wasn’t what they hoped. Maybe they didn’t want change but have to deal with it anyway. Maybe they didn’t particularly want this change, but they’re opportunistic enough to try and make it work for them in order to pursue their original goal. Whatever. What’s important is that the new circumstances start bringing those two motivations from earlier into conflict. When the players break up a drug smuggling ring the ninjas are running, they fight the cop’s ninja gangster brother, and the cop recognizes him during their debriefing afterwards. It’s worth noting here that circumstances had already changed in the world at large by the time the players even met the cop – his brother was already an evil ninja. What matters is that now he knows about it. Don’t feel the need to drag an arc out by starting it before the brother has even joined the ninjas.
Step three is developing the conflict. During this step, we explore the implications of the conflict. This can last just one scene for a one-adventure arc or several complete adventures for a campaign-spanning arc. What matters is that we explore potential answers and solutions to the conflict. This gives the players a chance to influence the arc NPC towards one answer or another. For our cop/ninja brothers, the cop brother might decide to let his brother go instead of letting his coworkers (perhaps the players) arrest him, whereupon he will undoubtedly face life in prison, or he might just use his position to focus on gang operations his brother’s not involved in. After all, the police have limited resources and can only stop so many crimes, is it really a big deal if he decides to leave one specific criminal permanently at the bottom of a most wanted list he knows he’ll never get all the way through? How does the ninja brother react to the knowledge that he has someone willing to look the other way for him, and just how far will the cop brother let his ninja brother go when he realizes that the ninja is exploiting his familial affection to get away with bigger crimes? Particularly since, from the ninja’s perspective, it’s perfectly reciprocal – he’s making sure that the ninja gang doesn’t hurt any of his own family members, after all. Maybe even directing them away from killing the cop brother when the ninja gangsters decide to retaliate against a crackdown by going on a cop killing spree.
What’s important is not that good guys and bad guys are fighting, but rather that different complications are being added to the basic conflict between the two motivations – love for his family and duty to the law – that drive our arc NPC. Having bad guys to fight helps keep players engaged and makes use of what is often the most developed system of an RPG, so that is definitely a plus, but it’s not the goal. What’s important is that players have a chance to influence the NPC as they react to the evolving shape and stakes of their internal conflict.
Ultimately, there is a turning point. It becomes completely impossible for the NPC to hold onto both their conflicting motivations. They must make a final decision about which one is more important. The ninja gangsters are planning to assassinate the chief and get him replaced with an easily bribed/intimidated crony, the whole department is about to go crooked, and the ninja brother has arranged for his cop brother to be amongst the crooks. All he has to do is not prevent the assassination. For maximum drama, have him (and the players) learn about the assassination from other ninja gangsters as the ninja brother is already on his way to carry it out. At this point, the cop brother can either head out to stop the assassination and bring his ninja brother in (betraying, according to the ninja brother, the trust between them not to hurt one another despite their conflicting roles in society) or else not do that and accept a new role as crooked cop helping his ninja brother evade the law.
For maximum interactivity, have the assassination go wrong (possibly because the PCs and cop brother prevented them before they knew the ninja brother had been tasked with the night’s big hit) and a bunch of the other assassination targets are on their way to back up the chief. Now, instead of just having to sit back and do nothing (a boring option and therefore one the PCs will be directed strongly away from), the PCs are asked to make a choice between intercepting the police back-up or preventing the assassination. Depending on their interactions with the cop brother so far, he might follow them whatever they do or pick one side or the other regardless of whether the PCs get cold feet at the last minute, but what he does should flow from how the PCs have directed him during the conflict’s development. If they constantly pushed him to let the ninja brother go and only now that the stakes are higher do they decide to about-face on that, the cop brother might not be willing to turn around on things like that. If they’ve been trying to shut these ninjas down, no excuses, but the offer of tons of money and power convinces them otherwise, the cop brother might continue pursuing the same duty they’d encouraged him to before – even if the PCs did turn out not to actually care about the law in the end.
Finally, there is the ultimate confrontation. There is a climactic showdown in which the NPC has committed to rejecting one motivation in favor of the other, and must triumph over some incarnation of that rejected motivation. For our cop/ninja example, this is obviously the cop brother either fighting his brother to prevent the chief’s assassination or fighting his honest coworkers to protect his brother. The NPC is generally going to come across as a lot more driven and badass in that phase than earlier, because they are no longer held back by their conflict and are fully committed to a specific course of action. In fantasy or sci-fi stories, this is sometimes accompanied by some kind of magic or techno upgrade or limit break, but that’s not as good an idea for TTRPGs as it is for books or movies. You still don’t want the NPC to overshadow the PCs in the final confrontation, and instead limit the third act power up to more of a metaphorical power derived from commitment and a willingness to end things once and for all rather than a literal and measurable increase in combat prowess.