(Spoilers for The Devil Wears Prada, Avatar the Last Airbender, Kung Fu Panda 2, and The Hunger Games triology).
Writing antagonists and villains can be hard, especially if you don't know how to do so.
I think a lot of writers' first impulse is to start off with a placeholder antagonist, only to find that this character ends up falling flat. They finish their story only for readers to find the antagonist is not scary or threatening at all.
Often the default reaction to this is to focus on making the antagonist meaner, badder, or scarier in whatever way they can- or alternatively they introduce a Tragic Backstory to make them seem broken and sympathetic. Often, this ends up having the exact opposite effect. Instead of a compelling and genuinely terrifying villain, the writer ends up with a Big Bad Edge Lord who the reader just straight up does not care about, or actively rolls their eyes at (I'm looking at you, Marvel).
What makes an antagonist or villain intimidating is not the sheer power they hold, but the personal or existential threat they pose to the protagonist. Meaning, their strength as a character comes from how they tie into the themes of the story.
To show what I mean, here's four examples of the thematic roles an antagonist can serve:
The Devil Wears Prada
Miranda Priestly is initially presented as a terrible boss- which she is- but as the movie goes on, we get to see her in a new light. We see her as an bonafide expert in her field, and a professional woman who is incredible at what she does. We even begin to see her personal struggles behind the scenes, where it’s clear her success has come at a huge personal cost. Her marriages fall apart, she spends ever waking moment working, and because she’s a woman in the corporate world, people are constantly trying to tear her down.
The climax of the movie, and the moment that leaves the viewer most disturbed, does not feature Miranda abusing Andy worse than ever before, but praising her. Specifically, she praises her by saying “I see a great deal of myself in you.” Here, we realize that, like Miranda, Andy has put her job and her career before everything else that she cares about, and has been slowly sacrificing everything about herself just to keep it. While Andy's actions are still a far cry from Miranda's sadistic and abusive managerial style, it's similar enough to recognize that if she continues down her path, she will likely end up turning into Miranda.
In the movie's resolution, Andy does not defeat Miranda by impressing her or proving her wrong (she already did that around the half way mark). Instead, she rejects the values and ideals that her toxic workplace has been forcing on her, and chooses to leave it all behind.
Avatar: The Last Airbender
Fire Lord Ozai is a Big Bad Baddie without much depth or redemptive qualities. Normally this makes for a bad antagonist (and it's probably the reason Ozai has very little screen time compared to his children), but in Avatar: The Last Airbender, it works.
Why?
Because his very existence is a threat to Aang's values of nonviolence and forgiveness.
Fire Lord Ozai cannot be reasoned with. He plans to conquer and burn down the world, and for most of the story, it seems that the only way to stop him is to kill him, which goes against everything Aang stands for. Whether or not Aang could beat the Fire Lord was never really in question, at least for any adults watching the show. The real tension of the final season came from whether Aang could defeat the Fire Lord without sacrificing the ideals he inherited from the nomads; i.e. whether he could fulfill the role of the Avatar while remaining true to himself and his culture.
In the end, he manages to find a way: he defeats the Fire Lord not by killing him, but by stripping him of his powers.
Kung Fu Panda 2
Kung Fu Panda 2 is about Po's quest for inner peace, and the villain, Lord Shen, symbolizes everything that's standing in his way.
Po and Lord Shen have very different stories that share one thing in common: they both cannot let go of the past. Lord Shen is obsessed with proving his parents wrong and getting vengeance by conquering all of China. Po is struggling to come to terms with the fact that he is adopted and is desperate to figure out who he is and why he ended up left in a box of radishes as a baby.
Lord Shen symbolizes Po's inner struggle in two main ways: one, he was the source of the tragedy that separated him from his parents, and two, he reinforces Po's negative assumptions about himself. When Po realizes that Lord Shen knows about his past and confronts him, Lord Shen immediately tells Po exactly what he's afraid of hearing: that his parents abandoned him because they didn't love him. Po and the Furious Five struggle to beat Shen not because he's powerful, but because Po can't let go of the past, and this causes him to repeatedly freeze up in battle, which Shen uses to his advantage.
Po overcomes Shen when he does the one thing Shen is incapable of: he lets go of the past and finds inner peace. Po comes to terms with his tragic past and recognizes that it does not define him, while Shen holds on to his obsession of defying his fate, which ultimately leads to his downfall.
The Hunger Games
We don't really see President Snow do all that much on his own. Most of the direct conflict that Katniss faces, but with his underlings and the larger Capitol government. The few interactions we see between her and President Snow are mainly the two of them talking, and this is where we see the kind of threat he poses.
President Snow never lies to Katniss, not even once, and this is the true genius behind his character. He doesn't have to lie to or deceive Katniss, because the truth is enough to keep her complicit.
Katniss knows that fighting Snow and the Capital will lead to total war and destruction- the kind where there are survivors, but no winners. Snow tells her to imagine thousands upon thousands of her people dead, and that's exactly what happens. The entirety of District 12 gets bombed to ashes, Peeta gets brainwashed and turned into a human weapon, and her sister Prim, the very person she set out to protect at the beginning of the story, dies just before the Capitol's surrender. The districts won, but at a devastating cost.
Even after President Snow is captured and put up for execution, he continues to hurt Katniss by telling her the truth. He tells her that the bombs that killed her sister Prim were not sent by him, but by the people on her side. He brings to her attention that the rebellion she's been fighting for might just implement a regime just as oppressive and brutal as the one they overthrew and he's right.
In the end, Katniss is not the one to kill President Snow. She passes up her one chance to kill him to take down the new threat of President Coin.