I think there are different types of villains to be considered in this respect. In brief, how I would interpret them:
The Madman
This villain is in it, to borrow a colloquial expression, for the lulz. They'll murder, cheat, steal and torture and won't think twice about stabbing somebody in the eye socket because it seems funny at the time. Utterly devoid of morals and crazy to a terrifying degree, they laugh at the hero's misfortune and treat everything like one big joke that only they seem to get. They love what they do and do it because they find wanton murder and chaos to be their idea of a good time. They exist to cause trouble and they live to make things that little bit bloodier and more amusing for themselves. In spite of their heinous deeds, they're eminently charismatic and you can't help but break into a smile and laugh along with them, even as they set fire to orphanages and drown puppies for the hell of it. Bad to the bone they may be, but they've got style, and they're just so much fun to watch.
See: Hades, Metal Face
The Dark Lord
The epitome of the term "big bad", this villain is cold, ruthless, rational and utterly awe-inspiring. They inspire fear and respect in all that behold them, and none dare stand before their overwhelming might: those that do are trampled underfoot and ground to dust beneath their heel, and woe betide anybody who thinks they can challenge their godly fury and magnificent power. They usually have a desire to rule their respective world and will stop at nothing to achieve their goals - such petty concepts as good and evil have no place in their worldview, a perspective borne of blood and tempered with flames of the darkest black. Their will is unbendable, their dedication unfaltering; this villain is the true lord of darkness, a foe worthy of standing proud as the greatest threat to the hero and manifesting the concept of a final boss.
See: Ganondorf, Nergal
The Tragedian
A villain of circumstance rather than choice, this villain has no desire to be the bad guy, but fate has other plans for them. Thrust into a role not of their own volition, but of the cruel whims of destiny, they must carry out their part in a deadly narrative and meet their untimely end at the hand of the hero. And for what purpose? Nobody can say for sure; this villain is just a pawn of their own unfortunate setting, a tragic entity forced to do evil things for reasons that they did not choose nor want to carry out in the first instance. They may be drawn into a terrible situation against their will and have to do evil things to make the world right again, or they may be so corrupted by hatred and malice that the darkness is all they know. Either way, this is a villain that, in another world, could have been a hero or a friend - a villain that you will no doubt feel just a little bit sorry for killing, and whose soul-stirring tears and scorned tale will make you wonder if there could have been any other way.
See: Count Bleck, Mewtwo
The Psycho
What's so different between this villain and The Madman? Well, for starters, this villain isn't funny - and if they are, it's not in the ironic, self-conscious way that The Madman is. This villain is vindictive, merciless, malicious and cruel - and that's just their job description! This villain's motivation is simple: hate. They hate the hero, they hate everything they stand for, they hate them for ruining their schemes and they hate them just for being them. They will stop at nothing to destroy the hero, to tear them apart, to make their life a living hell and to watch them suffer for their own demented sense of pleasure. This villain is just as prone to theatrics and flamboyance as The Madman, of course, and sometimes you'll even find yourself laughing at their incessantly cruel nature; but really, it's more of a nervous laugh, a brief moment of shock and awe at just how singularly insane and ruthlessly devoted they can be. Whereas The Madman thinks the world would be a better place if there were more blood and guts being spilled by their own wicked hand, this villain doesn't care about causing chaos for its own sake and doesn't want others to laugh at them: they just want to kill the hero, and damn anything that gets in their way. This villain is cruel and utterly, insanely dedicated to that cruelty, and when they set their mind on something, you'd better hope it's not you they're coming after.
See: K. Rool, Ridley
The Bully
This villain is...well, not really much of a villain at all, honestly. They're more so a villain in spirit than anything else: a bit boisterous, a bit rude, they may even try to take over the world or kidnap a princess or two - but they're not really serious about it. Well, okay, maybe they're serious about it - but take one look at their bumbling antics and hilarious shenanigans and you'll forget that they're supposed to be the bad guy. Sometimes, if an even greater threat looms before them, they'll even form a temporary alliance with the hero to save the day. Of course, it's straight back to business as usual after that, but it at least demonstrates that they're not so bad after all. Don't let the clumsy demeanour and silly gags deceive you, though: this villain can be quite deadly when they want to be, and just because they keep getting beaten doesn't mean they're not capable of holding their own or that they won't be back. What they lack in common sense, they more than make up for in tenacity and determination. Win or lose, you've not seen the last of them: they'll be ready and waiting next time with some even bigger plan and even wilder scheme, and they won't stop until they've finally proved themselves as the top villain.
See: Bowser, King Dedede
The Rival
The least villainous of all the different types, this villain isn't actually a villain at all. They're antagonistic and oppose the hero, sure, but that's just because it's their job: they'll fight the hero to the depths of hell and back, but they'd just as easily come to their aid if they had to. More of an anti-hero than anything else, they oppose the hero not out of hatred or malice, but because it's convenient or because it's more fun that way. More often than not, they really do want the hero to succeed, and they see their constant clashes as a way of training them and helping them carry out their noble deeds. Even if they do really want to beat the hero, it's nothing personal: they're irrevocably intertwined, you can't have one without the other, and when push comes to shove, who knows you better than your rival?
See: Wolf O'Donnell, Blue
