As Earth's Man of Steel, Superman is a DC superhero that is virtually indestructible. He's able to fly, has super strength, laser vision, X-ray vision, and is pretty much capable of withstanding any physical punishment thrown at him. Except for Kryptonite, his greatest weakness, he's a hero that has fought and beaten some of the most dangerous supervillains in his universe, thanks to his superhuman abilities.
However strong and resourceful Superman is, there are still some villains in the DC universe who have posed significant threats to Superman's well-being, sometimes nearly killing the Man of Steel in a fight--and once even accomplishing that historic feat in the famous "Death of Superman" storyline. These villains possess powers equal to or stronger than Superman's own, utilize Kryptonite in some crucial way, or use their intelligence to ruthlessly try to destroy Superman, once and for all.
Metallo
A cyborg that blends a robotic, android exoskeleton powered by a human brain and fueled by kryptonite, Metallo is essentially DC's version of a kryptonite-powered Terminator, out to rid Metropolis of Superman.
Metallo's primary power is super-strength, but he also has the unique ability to absorb any metallic surface into his body, making it an extension of himself. Normally, any regular cyborg wouldn't pose much of a threat to Superman, but Metallo's unique integration of kryptonite makes him a villain both fans and Superman might underestimate, but who can be an incredibly dangerous enemy.
Bizarro
One of the most popular villains in DC comics, Bizarro is the complete antithesis of the Man of Steel. Hailing from Bizarro World--an alternate world that is home to numerous strange, alternative characters like Batzarro, the World's Worst Detective--Bizarro is essentially an evil mirror version of Superman.
Portrayed equally as a comedic villain and tragic ant-hero, Bizarro isn't necessarily along the same lines as Darkseid or Lex Luthor in terms of pure evil. He's even shown to be an unlikely friend to Superman occasionally, known for being a confused, less-than-intelligent villain with a tendency to punch first, think later. However, because he possesses virtually the same powers as Superman, he can still be an incredibly powerful and dangerous villain to combat.
General Zod
As a fellow Kryptonian to Superman, General Zod--like Bizarro--has identical powers to Superman. However, unlike Bizarro, who has been shown to feel remorse and guilt for his actions, Zod is a far more formidable enemy, a cold, tactical leader with a severe God complex.
One of Superman's most enduring and well-known villains, Zod's appearance in comics has undergone some physical changes over the years, but he's almost always shown to be an aggressive, strategic man with plans to create a new world order. His main goal has been to turn Earth into his twisted version of Krypton, with the General positioned at the top of this new world he plans on creating.
Cyborg Superman
Another evil version of Superman, Hank Henshaw was a NASA astronaut using Kryptonian technology to impersonate Superman after the real Man of Steel's death.
Henshaw's original goal was to use Superman's technology against him, though his plans for revenge were put on hold when he learned Doomsday had already killed Superman while Henshaw was off, developing his plans. Without Superman there to stop him, Henshaw wreaked all kinds of chaos in the world, including causing Mongul's ship to crash into Coast City, killing millions of civilians, as a result, turning Hal Jordan--the original Green Lantern who was a native of the city--into the villainous Parallax.
Parasite
A dangerous, supervillainous leech with the power to drain other heroes and villains of their superpowers and absorb them as his own, Parasite is a key Superman adversary for this very specific reason.
Like most comic characters, Parasite's backstory has changed over the years, but his powers more or less always remain the same. Additionally, not only does he have the ability to absorb superpowers, but can also absorb information, knowledge, and even energy from other beings, killing them with a single touch. He's an incredibly underrated Superman villain, and one of the most deadly as well.
Brainiac
A cold, unfeeling, analytical alien android/cyborg, Brainiac is Superman's most intelligent supervillain, capable of outthinking the most gifted of Earth's minds--such as Lex Luthor, who has even managed to use as a pawn against Superman in a few comic book storylines.
More machine than extraterrestrial being, Brainiac lacks virtually any humane emotions such as mercy or sympathy, making him an unmoving villain capable of committing horribles without even the slightest hint of guilt. In the past, he's managed to shrink Krypton's capital city of Kandor, stealing it for himself, and in some continuities is even responsible for destroying Superman's homeworld.
Mr. Mxyzptlk
Don't underestimate his puckish physical appearance. Mr. Mxyzptlk may initially appear to be one of the lamest Superman villains there is, but hidden behind his fairly ridiculous wardrobe choices lays a being with infinite powers and abilities, virtually a god in human form.
In Superman's rogues' gallery of villains, Mr. Mxyzptlk is equivalent to that of Marvel's Loki--a fun-loving prankster who uses his vast powers for a good laugh, usually at other heroes' expense. Though Mr. Mxyzptlk may be a more trickster than a traditional supervillain, his powers are virtually beyond comprehension, able to warp reality in any way he sees fit.
Doomsday
One of the few supervillains who has killed Superman, Doomsday made an immediate impact shortly after his comic book debut, going on to be one of the most physically imposing villains in the entire DC universe, accomplishing a feat no other villain before him had ever accomplished.
A prehistoric Kryptonian, Doomsday is genetically mutated to adapt to the most punishing of physical blows, regenerating stronger than ever every time he is knocked down. An immortal being with jagged bone protrusions that he uses as weapons, he does not eat or sleep and exists with the sole purpose to destroy.
Lex Luthor
Superman's arch-enemy and a villain arguably as well-known as the Joker (the other mainstay DC villain), Lex Luthor is a megalomaniacal genius who believes it is his destiny to save the world from the threat of Superman.
His cause to check Superman's unlimited power may be "noble," but the lengths Luthor goes to destroy the Man of Steel make him an even bigger threat than Superman. Luthor's vast intelligence is his primary weapon, something he uses to elevate himself into high positions of power or build various weapons capable of killing the Kryptonian. His twisted worldview and hopes to one day become the new Superman--a being that Earthlings view as a god--make him one of the deluded, unstable villains in the DC universe.
Darkseid
Darkseid may be more along the lines of stereotypically pure evil incarnate in Superman comics, but it's for that very reason he remains the most deadly villain in the entire DC universe—someone who takes the entire Justice League coming together to defeat.
An alien warlord on the planet Apokolips, Darkseid is a powerful dictator who plans to enslave the entire multiverse (literally hundreds of billions of people), eliminating anyone who resists. The archenemy of the Justice League, Darkseid has a huge amount of powers and resources at his disposal, including his superhuman abilities (flight, strength, telekinesis, and his signature Omega Beam laser vision), and his army of militant Parademons.