Green Goblin started as a usual villain for Spider-Man but nothing really special that set him apart from the rest of Spider-Man's villains... Until he figured out Spider-Man's secret identity and was revealed to be his best friend's father, Norman Osborn. Norman used Spider-Man's alter ego to try and attack the ones he is closest to and this leads to the most iconic death in Marvel Comics, the death of Gwen Stacey. After kidnapping and "sort-of" causing the death Spider-Man's true love, Green Goblin mercilessly taunts Spider-Man for accidentally killing Gwen Stacey when he tried to catch her from falling off the bridge.
After a big climatic battle, Spider-Man finds himself unable to get revenge and kill the goblin and just let the Police deal with him. However, the scars that Green Goblin gave to Spider-Man is something that only a great arch-enemy can do.
The Incredible Hulk/The Leader
The Incredible Hulk is a large, mind-less behemoth of destruction while The Leader is a mastermind of strategy and evil. Samuel Sterns started as a everyday laborer in a chemical plant until a freak accident with gamma radiation that turned his skin green and grew his head, filling it with an extreme amount of knowledge. After he started losing his mental abilities, he made a deal with Hulk that he could make him Hulk forever without having to revert back to puny Banner. The deal was done when Leader took all the radiation from Rick Jones and made him more mutated and radiated than ever.
In retaliation, Leader broke their deal and dropped a gamma bomb on a town in Arizona; killing 5000 innocent people in process and "seemingly" kills Hulk as well. There is much more to the story but it's way too long to go into more detail. The Leader is a great opposite of the mean green machine.
Thor and Loki grew up together under the parental guidance of Odin and Frigga. Odin raised both the boys in believing that one of them would take over the throne of Asgard when Odin passed on. However, Loki found out that he was adopted by Odin when he was a infant and that he is actually a frost giant. Odin always meant for Thor to take over the throne because he would never have a Frost Giant in charge of Asgard, a species that has been in war with Asgardians. This made Loki increasingly more jealous and vengeful of both Odin and especially Thor.
Loki would constantly try to take over the throne and off his brother and eventually landed himself a banishment from Asgard. This made him even more hateful of his brother and quickly started longtime rivalry between the estranged brothers.
Yes, apparently Ultron is not Hank Pym's archenemy. Elihas Starr was a government worker who got himself jailed after smuggling secret blueprints and attempting to sell them on the black market. The media dubbed him "Egghead" and he was contacted by a crime lord who wanted him to prove his worth by defeating the great Ant-Man. Time and time again, Egghead would try to take down Hank but would constantly be met with failure. Egghead soon regarded Hank has his arch-nemesis and would keep obsessing over his intellect being inferior to Hank's.
After losing his father in an accident, Johnny Blaze was adopted by the Simpson family where he looked up to his new father-figure, a professional stunt man named Crash Simpson. He also developed a closeness with Crash's daughter Roxanne. Crash developed terminal cancer when Johnny was 19 years old and wasn't given much time to live. Johnny was then met by an evil demon named Mephisto who made a deal with Johnny so that Crash's life would be sparred of the cancer and Mephisto would get Johnny's soul. Johnny agreed to the deal and Crash was cured of his cancer... Only for him to die in a stunt accident. Having been tricked, Mephisto attempts to collects Johnny's soul but Roxanne intervenes and the love between the two made it impossible for Mephisto to take Blaze's soul. (Yes, the devil was defeated by the power of love)
In retaliation, Mephisto bonded the demon Zaranthos to Blaze's soul and turned him into the Spirit of Vengeance. Mephisto has been trying to use Ghost Rider ever since to do his evil bidding and control the Rider.
During a battle with The Punisher, Billy "The Beaut" Russo was thrown through a glass window and was thought to be dead. He survived and was horribly scarred across his face. He took the name nickname "Jigsaw" and took a strong vendetta against the Punisher. Jigsaw started his vendetta by trying to simply destroy his life and kill him in the process. However, after each failed attempt, Jigsaw soon became dangerously obsessed with Punisher and felt that Punisher was almost an important part of his life.
What's the best way to torture a man who is clearly insane... Steal what little identity he has left. After a violent spat at the Hellhouse with T-Ray, Deadpool wakes up from a coma in the snow after several days. He then finds T-Ray telling stories of his mercenary adventures that sound suspiciously familiar to Deadpool and that he was the real Wade Wilson. However, Deadpool actually starts to believe the stories and deeper into insanity. Soon, though, Deadpool decides to brush off the stories because he didn't care if they were true or not because he was just trying to be a better person. This caused T-Ray to be filled with hatred and bitterness toward Deadpool and start a long-time rivalry.
At one point, Moon Knight and Raoul Bushman were allies and even friends working together as mercenaries in the Sudan. When a doctor discovered a large amount of gold, Bushman tried to beat information out of him and his daughter and kill them. Marc Spector (Moon Knight) was disgusted by Raoul's actions and attempted to stop him. Bushman ended up beating him within an inch of his life and leaving him for dead. This lead to Spector being saved by Khonshu, adopting the name Moon Knight and getting revenge on Bushman for the horrible things he's committed.
Blade's mother was having complications during her pregnancy and was going into labor. In a fit, she went to a doctor named Deacon Frost who turned out to be a vampire and feasted on Blade's mother during his birth. This made Blade into a half-vampire known as "The Daywalker" where he has all the abilities of a vampire but sunlight does not affect him. Blade then spends many years tracking down Frost to get revenge for the death of his mother.
Fantastic Four/Doctor Doom
Dr. Reed Richards and Victor Von Doom were once friends who quickly turned into rivals in University. One day, Victor thought that he found a way to recover his mother's soul from Mephisto (another long story) and Richards tried to warn him that it wasn't gonna work. Victor did not listen and as a result the experiment backfired, caused his face to be scarred and he was expelled from the University. This caused Victor to form a bitter hatred for Reed and soon his entire team of superheroes.
Jack Murdock was a down and out boxer who had a child to support for by himself. This caused him to become an enforcer for Wilson Fisk, the Kingpin of Crime. Soon, however, his son Matt discovered what he did for a living, felt terribly disappointed and ran away. While running, Matt was almost hit by a trunk carrying barrels of toxic substances and it got in his eyes, "seemingly blinding" him. Jack finds out that Fisk was responsible for transporting the toxic waste and wanted revenge. Although, Fisk found out about this and had him killed on the street outside his apartment.
Matt used this experience to aspire to become a lawyer and fight for the law as Daredevil and eventually fight for the freedom of Hell's Kitchen from the clutches of the Kingpin of Crime.
Captain America/Red Skull
Probably the oldest rivalry in all of Marvel comics, this duo has been at each other necks since WWII. Hitler funded an experiment to turn Nazi General, Johann Schmit into a super soldier under the brains of Dr. Abraham Erskine. The experiment was a success... To a certain point. Johann suffered terrible side effects including his face turning into a red skull and evil intentions becoming something that even Hitler feared. This lead to Johann taking the name Red Skull and starting his own Nazi regiment called "HYDRA". This brought him into contact with Captain America who went under the same experiment and only made him more honorable and good, something the Red Skull hates deeply
Crossfire is a cybernetic assassin who first tries to hire a group of agents to kill Hawkeye so that he could attack the superhero community that would attend his funeral. This, of course, failed and led to Crossfire being thwarted by Hawkeye. Crossfire soon grew a deep hatred for Hawkeye and his then partner Mockingbird.
One night, a man named Khan found the remains of a spaceship that belonged to a species called Makluan. On that ship he found some special technology and ten powerful rings. Khan would go on to find others who wanted to know the secret behind these rings and eventually started funding a war-lord by the name of Wong Chu who would eventually kidnap Anthony Stark and would set him on his path to become Iron Man.
X-Men/Brotherhood of Mutants
This bitter rivalry goes all the way back to when Charles Xavier and Erik Lensherr were once friends. Although they both had to deal with the prejudice of being mutants but Charles believed that humans and mutants could live in harmony if mutants could prove their worth to society. Erik, however, believed that humans were too far gone to change their beliefs and that he needed to rule over the humans to be accepted. This lead to bitter long rivalry that made them both start groups that followed their own beliefs and goals.
Not much that really sets these two apart for being archenemies expect that Chemistro would constantly try to defeat the Heroes for Hire that included both Iron Fist and Luke Cage. After being constantly stopped and taunted by Luke Cage, Chemistro made it his mission to take him down
Having been planted with countless memories that he doesn't know are real or not, he had a memory of Wolverine and decided to track him down to Canada. There, he raped and seemingly murdered his then girlfriend Silverfox. This enraged Wolverine to get revenge on Sabretooth.
War Machine/The Advisor
Not much is known about the Advisor except that he is a known terrorist and only known as War Machine's archnemesis.
Jessica Jones/Purple Man
Purple Man is considered to be a very sick and disturbed individual. Purple Man got his eye on Jessica Jones a.k.a Jewel and he used his mind control powers to torture her for countless months. Purple Man eventually controls her to go and kill Daredevil but she mistakes Scarlett Witch as Daredevil and get's beaten into a coma. The Avengers nursed her back to health and she wanted vengeance against Purple Man.
Dr. Ulysses Klaw created technology that could use pure sound as a weapon. However, he needed vibranium for it to work and went to Wakanda to find some. When the villagers were less than cooperative with him stealing their fortunes, he started slaughtering them and taking what he wanted by force. This made the people of Wakanda go to war with Klaw's forces and resulted in T'Challa removing one of Klaw's hands with one of his sound weapons. Years later, Klaw tried another attack where this time he could convert living creature into sound monsters. T'Challa (now Black Panther) was able to defeat Klaw once again with the help of Fantastic Four but Klaw still hungered for more precious vibranium.
The Sentry/The Void
Having suffered memory loss, Bob Reynolds remembered one day that he was a mighty superhero named Sentry and that the Earth was soon gonna be under attack by his arch-nemesis, Void. He summoned other great heroes to fight along side him. Everyone, however, is stunned that he has so little memory of his superhero past. Until it is revealed that he suffers from a split-personality where The Void constantly tries to take over Sentry's entire psyche. Sentry erased his own memory so that he could destroy Void in the process, a plan that did not work entirely.
Dormammu wants one thing and one thing only, to rule the all the realms while Doctor Strange has made it his sworn duty to protect the Earth dimension. Dormammu wants to rule like a dictator while Strange will sacrifice his own life for the innocent. Cornerstone to any great archenemy is to demonstrate how mirror opposite they are. Dormammu has little empathy for the weak and defenseless while Strange sometimes cares a little too much. Dormammu seeks to conquer the realms with an iron fist while Doctor Strange wants to protect them and keep the innocent free.
Titania was recruited by Dr. Doom to fight for him during the original secret wars. Titania took on She-Hulk as they both raged on about being the strongest women in the known universe. This is ultimately what started their long feud as they constantly try to out muscle each other to prove who is more powerful.
Namor: The Sub-Mariner/Attuma
While Namor can be quite a prejudice dick at times, he is a thoughtful and caring leader over his people of Atlantis. However, Attuma believes that he is the true ruler of Atlantis and he will stop at nothing to claim the throne, even if that means turning to violence, cruelty and even the harm of his own people he would swear to rule over and protect.
Iron Fist/Master Khan
Master Khan is a master sorcerer who has grown a bitter hatred for Iron First due to him always foiling Khan's evil plans. In a plan of retaliation, Khan summoned the Super-Skrull to Earth (who was dying of a rare decease) and made a deal that if the Super Skrull kills Iron Fist, Khan will cure his decease. Khan then turned the Super-Skrull into the character Captain Hero and erased his memory. Captain Hero soon befriended Iron Fist but the Khan's magic wore off and Hero's memory was back.
Iron Fist was forced to kill his once close friend to save his own life and this would start Iron Fist's bitter hatred for Master Khan.