As a writer who makes devils and demons a primary part of my story, it's easy to fall into the trap, "well, they're evil because x," without really exploring the purpose. Extremely basic villains will be evil for one reason, while really interesting, complex villains will have different motives that all culminate in them doing what they want, even innately evil beings like demonic entities. True evil isn't born; it's manifested through circumstance. We can look to history's greatest villains and see that they didn't see themselves as the bad guy; they saw themselves as the ones who'd save their nation or ideology, thinking they were doing what was right. The most evil regimes to have existed had precursor causes that allowed for dictators to take power and not only oppress their own people, but to use others as a scapegoat and engage in sadistic evil beyond words.
The easiest but also one of the most memorable types of villains to write is the unexpected type: the cartoony, funny type of villain. There's a reason people talk about Doctor Doofenshmirtz from Phineas and Ferb, and while anyone can make the argument that he's this "supervillain", he's amusing because he has these elaborate inventions and plans he uses to try and get control over the tri-state area, often foiled by Perry the Platypus. Doofenshmirtz is an icon in our culture for his comedic role; despite being a "villain", he was goofy and entertaining. Depending on your wish for writing a villain, this might be a good villain to use for your story, and this is not the wrong way to write! It's your story.
As for the true evil villains? The ones I find haunting?
There's a reason people use the nazis as a stock enemy: it's because everyone can unanimously agree that nazism is the king of evil ideologies. The nazis are haunting villains not only because they actually existed, but because of what they were capable of: propaganda, workforces, soldiers, and engineers; they were capable of great evil and planned it out. Not every story will have harsh villains in it, and in fact, while many stories might include villains like nazis, they often fall short of their real-life equivalents of evil. Even still, despite the indisputable evil that they were, they would attempt to justify their evil as righteousness, as does any villain.
In many pieces of fiction, I can make the argument that the villains are typically more complex than the hero. Heroes tend to want to be righteous, save for others who later come to that conclusion (like Han Solo). With villains, you can grant them similar past circumstances to the hero, but the difference is how they may react or perceive their decisions. The difference between a hero and a villain's past can be whether or not they chose to give up being good, which automatically assumes that the villain is weaker than the hero. Beyond this, I think that a good villain is an inevitable villain. Going back to real history, Hitler was a painter in his youth and was rejected from art school not once, but twice. There's a superficial understanding in this historical context that "Hey, if Hitler was accepted to art school, there'd be no World War 2!" when the reality is that the geopolitical circumstances around the Treaty of Versailles essentially guaranteed that a Second World War would happen anyway. Maybe it wouldn't necessarily be the Nazis themselves that gained power, but chances are high that it'd be a similar revanchist-expansionist type ideology (Fascism started with Mussolini, and he got into power long before Hitler. Despite Italy being on the winning side of the First World War, they felt like they didn't get enough concessions that gave the Italians a man to venerate). Applying this to writing, historical conditions in your world indicate a villain's rise to power, their backstory, and how they might perceive the world around them. If they're forged by the flames and hatred of history, we have reason to suspect they'd be the most evil villains of all.
Moving to a more fictional villain that's still haunting, a lot of people who read literature will make the argument that Judge Holden from Cormac McCarthy's Blood Meridian is the most evil, haunting character ever written. Remember that this is not without substance: Judge Holden was a real person (who McCarthy fictionalized for his narrative), and Samuel Chamberlain wrote that "Not a crueller villain was left unhung", the man whom McCarthy would base his protagonist on (who actually met the real Judge Holden in a gang of scalphunters). Every crime possible, Judge Holden is guilty of, and while at face value that sounds like it's being made for shock factor, the point of Blood Meridian is that violence is perpetual, needless, and unnecessary. Even still, in the book, Judge Holden is haunting not just because of the evil he is capable of, but because of the good traits of humanity he has within him. He has a genius-level intelligence, is a skilled drawer, is genuinely interested in the world, and will bow to gentlemen and criminals alike. He spends the book trying to corrupt the protagonist and thinks he's capable of such.
A tamer villain than these men might be Gus Fring from Breaking Bad. He's a brilliant drug lord, but he remains cool, calculated, and compared to Walter White, you can make the argument that he's less evil. Gus is one of the most well-known TV show villains (props to Giancarlo Esposito for playing him so well!), and while watching the show, it's often easy to forget that he's a true villain, someone who's purely evil. It's because his motivations are human; he acts compassionately toward his employees and his business partners. He has violence as a last resort, and while that may not seem villainous, he is making millions of dollars selling methamphetamine illegally. The show does a great job of humanizing its villains, making them 3-dimensional and not just stock characters.
As for me, I love creating villains and figuring out huge obstacles for my protagonists. A good villain isn't just someone who challenges your protagonist; it's someone who mirrors them. I've spent plenty of time studying what makes evil, and it's truly a fascinating subject. In fact, it's not just villains, but sometimes just antagonists who share a different viewpoint from your protagonists. I don't consider any of my protagonists anti-heroes, but if you're writing them, they have villainous parts about them. It's also possible to write a villain protagonist, and so your antagonist is the hero, and you have to figure out how to "defame" from that point of view.
Villains have great versatility, and I think it's a common misconception about most villains. While anyone can agree that the movie protagonist killing 1,000 of the villain's henchmen, then hesitating at the final boss's throat, saying, "I'll be just as bad if I kill him!", a common gripe people have is the huge villainous plan. I think it depends more on how the plan is presented and the psychology of the villain. Some villains want to be talked out of their plan, and that's why they might talk to the hero about it. Some villains might also follow a code of honor, and that's why they don't kill the protagonist immediately, the first chance they get. Having a code of honor doesn't make them any less evil; it just makes them more human characters. Villains, like heroes, are fallible, and that's why they'll slip up and make mistakes like the hero. All in all, it's just a perspective on writing.
While the term "villain" varies based on what you're talking about, you can make the argument that villains are not exclusive to characters. They might be environmental, such as the weather or natural disasters. They might be a manifestation of an idea, and some might have never existed in the first place. The latter is a great tactic for writing horror (take your monster or whatever is scary out of your scene. Is your scene still scary? If it is, it's a good horror scene!). I'd suggest thinking of ideas of the type of story you want to tell before you figure out what type of villain you want to use for your story. A serious book might contain a hardcore villain with a 3-dimensional personality, whereas something more comedy-driven and lighthearted might have a silly villain. Either way, it's dependent on what you're writing.
The amusing villain is an interesting archetype and always will be. However, I think a true villain in terms of evil will always be those of historical evil. To know humanity is capable of such great evil and yet so many people are generally good is always a light in shining darkness.
Until we meet again, stay cool, everyone
Dustin