Superman is the modern Ideal Hero. Sure, plenty of people like to screw around with him, and the elseworld stories make Beware the Superman all the more appealing, but in the "modern" sense, Superman is the ideal superhero: wise, strong, compassionate, loving, a balanced life in both superheroing and his normal life, idealistic and able to win fights without resorting to lethal violence. For that reason, his villains usually take one of his ideal traits, and turn them on their head. Not like Batman per-se, but still somewhat similar.
Lex Luthor: Superman, despite being an alien, was raised to believe that Humans Are Special, and strives to help humanity however possible with little thought for reward. Batman even calls Clark "One of the most human of us all." He also believes in being accountable for all he does. Lex, on the other hand, is everything Superman at once hates and yet needs. On one hand, he's egotistical, prideful, greedy, manipulates what exists for his own benefit, and hates having anyone talk down to him. On the other, he's super intelligent, and would be exactly the kind of man that would change the world so that it wouldn't need a superman anymore. He's Humans Are the Real Monsters, but at the same time, embodies so much potential, that Superman is at once frustrated and distasteful towards Luthor.
Brainiac: Superman is the benevolent alien who took up with us. Brainiac is "everything bad about aliens", all rolled into a robotic body with a singular mission to collect and destroy things. Also, as Superman gets his powers from his unique biology, Brainac contrasts in another way for using technology to fight the man of steel.
Darkseid: Superman never puts himself above others, at least not in the tyrannical sense. Darkseid on the other hand lords over Apokalips as its Lawful Evil tyrant, and is so manipulative and cruel, that Superman actually hates him. He may be impassionate about going after Brainiac, but Darkseid is a being Superman actually hates.
Also the idea of Superman representing hope and Darkseid being fear both are ideas that each being wants to leave behind whether they are alive or not.
Bizzaro: (borrowed from fridge brilliance) He's an imperfect, backwards clone of Superman. He's also a metaphor for humanity on the whole. Think of it; Bizarro has all of Superman's powers (if reversed, in some continuities), and is strengthened by the Kryptonite Factor. In the same vein, humanity has surpassed most if not all of its natural boundaries, defying the elements time and again. So given this, why isn't Bizarro the hero? Simple: despite his power, he's not mentally capable of bearing the responsibility that comes with protecting the world. He's every bit as powerful as the Big Blue Boy Scout, but his twisted logic won't let him do the good and right thing despite how badly he wants to. Sounds about right.
General Zod: Everything Superman is (you know, as a Kryptonian), only Tyrannical instead of benevolent.
Mr. Mxyzptlk: The Oddball of Superman's Rogue's Gallery, an imp from the fifth dimension who likes to mess around with superman. However, this is actually perhaps the most hidden of foils, as Mxy is all of superman's power (metaphorically speaking since eelseworld tales have pretty much shown that if he wanted to, Mxy is more powerful than Superman), but none of the responsibility. He lives for his own amusement and mischief, rather than helping others as Superman does.
The Prankster and Toyman can probably fall under a similar category of difference, though they also share traits with Brainiac due to their use of technology to get what they want.
Parasite: Both feed off of strong energy sources. Superman from the sun, and Parasite from anything he can get his hands on.