Three types of story villains (and what they do and don't accomplish)
Starting off with the disclaimer here that I by no means think I'm the first person to tackle this concept: I'm spitballing based on my own lifetime of experience 1) storytelling and 2) interacting with readers and audiences.
So what kicked this off was that I've seen a lot of posts talking about the 'right' way to write villains -- recently, but also over the years -- and they always strike me as frustratingly incomplete. The recent ones especially tend to come with a strong dose of "you need to make sure that your villain is utterly detestable, with no redeeming or sympathetic qualities, and that they embody all the worst traits present in your work," etc, that sort of thing. And my immediate reaction to that is to say "well, maybe that's true for some villains, but it's not universal by any means."
So, if there's not just one way to write a villain character, how many ways are there? Lots, obviously, but based on my observations, the really affecting/iconic/impactful villains tend to fall into a few categories.
Villain Category 1: The Hate Sink
This is the sort of villain that the above advice is useful to: the guy whose role in the story is to be someone that everybody, both in and out of universe, hates. This guy needs to do horrible things, and just be horrible as a person and as an instigator of action, in order to build up a very satisfying catharsis when the hero defeats him. This is the guy (or girl) with the ultimate punchable face.
The primary emotion this type of villain inspires is anger. (Secondarily, contempt.)
Examples of this type of villain: Jin Guangshan, Trent Ikithon (tabletop,) any Leverage villain
Villain Category 2: The Power Fantasy
This is a big big broad category and a lot of subcategories are going to fall under it. This villain is one who is operating without rules or restrictions and at a high level of effectiveness. They are usually The Unfettered. It seems like there's nothing they can't do, if they feel like it; and while that makes them dangerous, it can also be very appealing. In terms of audience engagement this is the dinosaur: a toddler stomping around roaring pretending to be a T-Rex, because T-Rex doesn't have rules or bedtimes and can have whatever snacks they want! This type of villain is scary, dangerous and needs to be stopped, but they're also cool.
(They can also be a Power Fantasy to members of the audience that they share characteristics with: a queer-coded villain who doesn't give a fuck about societal mores can be very appealing to queer audience members who are closeted in real life, or a disabled villain who manages to crush the galaxy under their heel regardless of their disability is naturally going to be very appealing to audience members who share that disability, or a villain who goes on an unstoppable revenge rampage following an assault or some other trauma can feel very cathartic to an abuse survivor.)
The primary emotion this type of villain inspires is fear. (Secondarily, awe.)
Examples of this type of villain: Darth Vader
Villain Category 3: The Pain Mirror
This is the sort of guy I was talking about in my recent post about villain fans, villains whose role in the story is to take a trauma or tragedy and say to the world 'this pain matters.' This is almost always a Tragic Backstory villain, though it's possible to arrive there over the course of the story. This villain is usually someone going on a rampage in response to some awful event, but whose reaction to it threatens to destroy more than it can save and needs to be stopped thereby. This type of villain can be very impactful, but also very controversial depending on the handling.
Examples of this type of villain: Magneto
The primary emotion this type of villain inspires is grief. (Secondarily, pity.)
I'm sure there are more categories of villains, at least five; these three were just the ones I could come up with on a quick survey. And obviously (well, obvious to me anyway) there's going to be some amount of crossover; a Pain Mirror can also cover Power Fantasy territory, though there's not complete overlap, and a Power Fantasy that does enough despicable things can take on aspect of Hate Sink.
There's risk in letting these blend too much, though; a Pain Mirror who has too many aspects of Hate Sink threatens to become unsympathetic to the audience, and a Power Fantasy with too much Pain Mirror starts to look a little pathetic. But the biggest place I think some story villains run into trouble is if their creators haven't picked a category -- or can't or won't commit to one.
The example that came to mind when writing this post was Kylo Ren (sorry, Kylo Ren fans, if you're out there.) What type of villain is the movie trying to make him, exactly? He doesn't succeed at Power Fantasy because he can't actually accomplish any of his goals; he doesn't succeed at Pain Mirror because his pain is, quite frankly, self-inflicted and not relatable. He sort of slips towards Hate Sink by default, but it's clear that his writers wanted for the audience to find him cool and sympathetic. So he sort of accomplishes none of these?