A supervillain is a person with extraordinary powers who use their abilities for a variety of means detrimental to humanity. They often come into conflict with those known as superheroes.
There are a variety of ways a person can achieve superhuman powers.
Because of the rise in mutants, there are a great deal of people who are born with powers. Powers sometimes develop immediately at birth, but often at puberty.
It is also possible that someone born to superheroes, but not necessarily a mutant, can be born with powers. However, this would qualify them as a mutant.
In their quest to better understand the universe, scientist often make mistakes. Sometimes these mistakes give a person powers. Curt Connors accidentally turned himself into the Lizard when he wanted to regrow his arm. Leader fell into some radioactive waste. Green Goblin happened because of an accidental explosion.
Science on Purpose
Sometimes, a scientist will use their knowledge to create powers for themselves. Doctor Doom uses his scientific knowledge to try to rule the world. The Sentinels and Master Mold are creations of villainous humans.
There are some things science cannot explain, and some are able to use that. Vampires got their abilities through an ancient curse. Baron Mordo uses his mystic abilities to bring an evil entity to the world.
Not all villains come from Earth. Many come to the planet and abilities normal to them are extraordinary to the locals. Silver Surfer, Terrax, and Galactus came to Earth to destroy it. Mandarin got his powers when he found an alien space ship.
Not every supervillain necessarily has any extraordinary abilities. Some normal human being choose villainy despite having nothing to compete with more powerful beings. Red Skull has no notable abilities despite his gruesome appearance.
There are a number of villains who have no abilities of any kind but are able to antagonize heroes through other means. Graydon Creed Jr. and the Friends of Humanity are a terrorist organization. Kingpin controls a vast criminal empire. Justin Hammer has connections with The Hand. The Nazis were a powerful army.
There are a number of reasons one would become a supervillain.
Desire to Rule
In most cases, the villain simply wants to rule over others. Red Skull, Kingpin, and Doctor Doom are just trying to dominate the world.
Sometimes, a hero will be considered a villain to the public. The United States government hunts the Hulk, who often causes excessive damage in his heroics. The Daily Bugle often paints Spider-Man as a menace and often turns public opinion against him. The X-Men, and mutants in general, are demonized simply for being mutants.
Good Guy Gone Bad
There is the rare case when a hero will go bad. Spider-Man's mutation went out of control turning him into Man Spider, a large spider creature running on pure animal instinct. Jean Grey became hunted by the Shi'ar when the Phoenix Force almost destroyed an entire star system. Morph was brainwashed by Mister Sinister.
There are a number of people who do heroic deeds through not-so-heroic means. These are known as anti-heroes. They often come into conflict with other superheroes who object to their methods. They are not necessarily villains, but are antagonistic to other heroes. Punisher saves people but is known for excessive actions against others. Blade will kill any vampire without even trying to save them.
In various alternate universe, people make different decisions and the world turns out differently. Heroes sometimes become villains and villains sometimes become heroes. In the Age of Apocalypse, Magneto actually started the X-Men and fights for mutant equality. Meanwhile in one universe, Spider-Man is actually a villain as Spider-Carnage.