Azula, princess of the Fire Nation
One of my favorite villains is Azula from Avatar: The Last Airbender (the original animated series, not that monstrosity Shyamalan turned it into on the big screen). She's ruthless, cunning, manipulative, a firebending prodigy, and loves the power she has.
Azula in action!
Captain: Princess, I'm afraid the tides won't allow us to bring the ship into port before nightfall.
Azula: I'm sorry, Captain, but I do not know much about the tides. Can you explain something to me?
Captain: Of course.
Azula: Do the tides command this ship?
Captain: I'm afraid I don't understand.
Azula: You said "the tides would not allow us to bring the ship in." Do the tides command this ship?
Captain: No, Princess.
Azula: And if I were to have you thrown overboard, would the tides think twice about having you smashed against the rocky shore?
Captain: No, Princess.
Azula: Well, then, maybe you should worry less about the tides, who've already made up their mind about killing you, and worry more about me, who's still mulling it over...
Another villain fans seem to love (I don't, personally, but he is obscenely popular in the fandom) is Sephiroth from the game Final Fantasy VII.
Cloud and Sephiroth from the movie Final Fantasy: Advent Children
Sephiroth could be considered a sympathetic villain b/c of his background. But he becomes so undeniably evil that I can't say the sympathy lasts very long. Plus, he kills one of the game's characters and one of Cloud's potential love interests.
Both Azula and Sephiroth are still complicated villains. They wouldn't kick puppies just for fun--they would do it for specific reasons usually pertaining to hurting or manipulating their protagonist counterparts. And like sympathetic villains, they both have difficult pasts.
So what sets them apart? Intent. Motivation.
Azula controls people with fear because she doesn't trust anyone. In her own words, her own mother thought she was a monster. And it's sad and a bit tragic, but she also enjoys making people fear her. She enjoys the power. Sephiroth is a bit more complicated and it's hard to get into it without explaining his entire backstory. He didn't start out evil, but he ends up twisted and ruthless, someone who wants to "become a god that rules over the entire planet by merging with the planet life force, known as Lifestream, and taking control over it" (from Wiki).
These two are so good at being bad, that fans can't help loving them for it.
What do you think? Which kind of villain do you prefer? Loki or Sephiroth? Nuada or Azula?