I'll split my definitions of villains for this post, as I have different preferences for each type. I'll also disregard the backstory part as others have already talked about it. These are all personal preferences.
100% evil, you will never see them not be evil. They even eat evilly. They have the same personality whether with friends, enemies or third parties. Driven by a large, singular goal, usually world domination. Only interacts with the heroes to fight them.
Examples: Zordon, Fire lord Ozai, Kefka, Giovanni, Voldermort
-They focus on their goal with gusto
-To the point, does not taunt the protagonist unnecessarily
-Imposing, Scary figure that the protagonist dreads to face, but has no choice
-When they act as a stepping stone for the MC, as if their whole purpose is to train the MC. Examples include teaching the MC how to fight them ("Mwahaha, you cannot defeat me unless you have the sword of light")
-When they risk ruining their plan by doing something stupid like gloating or sparing the hero out of pity
You get to see samples of their lives besides being evil from the protagonist's point of view, through some form of exposition. A villain may be found to dote on his grandchildren, for example. Their shown personality is still the same though.
Examples: Gollum, Sephiroth (original FF7), Azula (Avatar), Thanos, Kronk, Hades (Hercules)
-Whatever extra information you learn about them is actually interesting
-Does not change the fact that they are villains
-Makes them more relatable
-No point to the revelation
-When they pretend to be 2D characters
Their perspective is shown to the player/viewer. They will interact differently with the heroes, their minions and other villains. They have an 'evil' persona they display and another for personal time. Most will have their own values or morals that might make them work with the heroes or fight other villains.
Examples: Team Rocket trio, League of Villains (BNHA), Dr Evil (Austin Powers), Bowser
-Contrasting personalities, maybe flamboyant to the heroes but actually shy
-When their own values get in the way of their goal, causing inner conflict
-Goofy evil-but-not character, e.g. Doofenshmirtz
-When they over compensate and suddenly become an anti-hero or full hero, and their personality changes without good reason. May be good character-wise but not good villains. (e.g. Deadpool)
-Flanderization that one personality is forgotten, and the villain becomes a joke character (e.g. Juggernaut from X-men)
Blurred lines between if they are really evil or not. Can be the protagonist in other stories. When players play as these characters, the heroes in turn becomes the Villain. Probably has a large focus in the story, and interacts a lot with the protagonist, may form alliance or truce.
Examples: Light (Death Note), Scar (FMA), Tom (Tom & Jerry), Pagan Min (Fay Cry 4), Every pirate in One Piece
-Fully sure of their values, how to obtain their goals and how the hero factors in
-Unpredictable to an extent, many factors go into their decisions
-Shows the similarities to the protagonist
-Manipulates the protagonist
-When the writers randomly throw in traits to 'deepen' the character (The Joker)
-Is stupid and easily tricked by the heroes
-Not actually a threat to the hero's goals, too weak to make a difference even with their deep values and morals
Chaotic evil villains are underrated, but also the number 1 choice for uninspired people which is the reason why they're underrated. If well written, you could absolutely get away with having a bad guy erase the hero's home town and leave saying "sue me, *****es".
There's just a difference between no chill and "muahaha I do bad stuff because I'm very evil!!!", so if you can't make sure you don't fall into the latter please do settle for a villain who isn't chaotic evil.
A good example of a good chaotic evil character is actually the player in the genocide route of Undertale. Why is Chara going out of their way to kill every single monster in the underworld? Because they can, because the player is curious of what would happen if they did. They have no moral motive, nothing they stand for that justifies it, just wanting to see what happens.
Contrary to most opinions, I like shallow evildoers.
I'm all about JUSTICE. And if I can relate to a villain, it does stop to be a villain, it become a commoner.
I want to mince, destroy, punish and banish pure evil. The mindless evil, demoniac evil, cthulhu evil, the one that does bad to people 'cause of his biological nature, not for a "drama". If someone got a reason for his deeds, he does stop to be "evil".
My idea of hero is one that protect humanity against such "alien" menaces. I'm more interested to explore the relation between hope, despair and courage of survivors than identify with a villain and his impossible reasons. Villains are just a plot tool to have humanity fell or shine in a crisis.
@CHKNRAVE, thank you for letting your voice be heard. @Dororo, thank you for letting your voice be heard.
I like a shallow evildoer too, but not only one villain in the game. I am sure people here going to get upset and say are you going mad. Other than that I like more cold calculate characters.
Remember Symbolic character is used to represent something larger and more important than themselves.
Kefka Palazzo represents the evil of humanity, Cloud of Darkness represents Chaos, and Sin monster in Final Fantasy X represents sin. I mean it is not complex because it is the personification of an idea.
I like a diverse cast of villains because I get uninterested quickly if things go too long. However, I do not know how many people like this idea walker.
For me, I like it when Villains aren't technically villains, or they don't think they are doing the wrong things. I also like villains who actually have a reason to be evil, like they were beaten down or have someone drive them to being the main antagonist
New PC is built. Now I walk that fine line between wanting it NOW, and being responsible and waiting for the stress test to be finished. By this time tomorrow it'll be up and running. And my desk will have had a much-needed clean.