There are many different types of villains in the world.
Villains started out as a group of people who wanted to use their new gifts to benefit themselves instead of helping others. Unlike heroes, these people wanted to use their Quirks to commit crime and evil. According to Naomasa Tsukauchi, in the current age, the crime rate committed by villains is quite low as a result of being pressurized by the huge amount of heroes.
A villains group called the League of Villains, led by Tomura Shigaraki, attacked the USJ in an attempt to kill All Might after infiltrating U.A. and learning of All Might's whereabouts. However, the majority of the League of Villains (except for Tomura, Kurogiri and Nomu) are defeated by Class 1-A. After All Might arrives, Nomu engages All Might in battle and is defeated. Tomura and Kurogiri try to finish what Nomu started, but their attempt is thwarted by Izuku Midoriya, and the timely arrival of the U.A. Pro Heroes. Both Tomura and Kurogiri are sucked into a Black Hole created by Thirteen (however, Kurogiri creates a portal back to their hideout, teleporting themselves out of harm's way). Thus, the League of Villains is defeated and the majority of the villains of the alliance (along with Nomu) are captured and detained by the police, leaving Tomura and Kurogiri as the only remnants of the League of Villains.
During the U.A. Sports Festival, the notorious villain called Stain heavily injured Ingenium, leaving him for dead. Afterwards on a rooftop, Kurogiri of the League of Villains asks Stain for a moment of his time and teleports him to the League of Villains' hideout.
At the League of Villains hideout, Tomura invites Stain to join the League of Villains as his notoriety would benefit them greatly. Tomura also tells Stain what their goals are. However, Stain is not amused with Tomura's child-like attitude, as well as his goals, and begins arguing with him. The mysterious voice on the monitor asks Kurogiri to let Tomura and Stain continue with their meeting as Tomura's meeting with Stain (an experienced infamous villain) will stimulate Tomura's growth.
League of Villains
Meta Liberation Army
Paranormal Liberation Front
Other Known Villains
Rankings according to My Hero Academia Official Character Book Ultra Archive which covered Villains from Chapter 1 to Chapter 88, and the updated rankings based on My Hero Academia Official Character Book 2 Ultra Analysis which covers up to Chapter 235.