Mewtwo's original name was Gigu-2 as a reference to the similar villain from Earthbound - Gieuge (better known as Giygas).
Mewtwo is based loosely off of Mew with feline features. Its body is purple and it has three fingers on each hand. Mewtwo has a faint purple Aura emerging from it at full power. Mewtwo was stated as the world's most powerful Pokemon to exist.
Mewtwo is highly adept at using its Psychic powers, displaying telekinesis and telepathy as well. It uses it's telepathy to communicate with humans and translate the Pokémon language. It is able to Fly and erase people's memories as seen in Mewtwo Strikes Back. Mewtwo when caught has a base happiness of 0, it is said to have the most savage heart of all Pokemon
In the anime
Mewtwo's anime debut was in Battle of the Badge! where Giovanni used it against Gary, who referred to it as an "evil" Pokémon. Mewtwo again appeared in It's Mr. Mime Time briefly. Mewtwo also appeared in Showdown at the Poké-Corral! where it escaped from Giovanni's gym, as seen in the movie. Mewtwo has been the focus of a few movies, being the main Pokémon inMewtwo Strikes Back, inMastermind of MiragePokémon under the control of Dr. Yung and Extremespeed Genesect and the Reawakening of Mewtwo. Mewtwo was also the star of the movie; Mewtwo Returns.
In the manga
It was created by Blaine in a laboratory and went on a rampage, almost destroying Cerulean City. When Red heard about "The Monster of Cerulean City", he decided to catch it. After a single encounter with the tornado that was Mewtwo, he met Blaine. Blaine told Red of his condition, how if Mewtwo continued to rampage, both Blaine and it would die. Blaine then continued to show Red his disfigured hand, a mark that he shared with Mewtwo, and concluded his talk with Red, declaring he intended to kill Mewtwo, in order to put it out of its misery, and also to save his own life. Red later encountered Blaine attempting to finish Mewtwo. Red, after stopping Blaine's attempted suicidal killing of Mewtwo, caught it using his whole team, and launching Pika with the Master Ball in its mouth, catching Mewtwo for Blaine. Blaine decided to train and raise Mewtwo as his own. Later, Mewtwo appeared alongside Blaine, battling a child helping them train. Blaine assisted Yellow from then on out, being paired with Yellow by Sabrina's Alakazam. In this setting, we learn that Mewtwo needs to rest in its Poké Ball, and if it was released for too long (over 5 minutes), it would die. It then appeared again in the FireRed and LeafGreen chapter, being loaned to Red by Blaine. It aided the Pokédex holders in stopping 9 Forretress from blowing up Neo Team Rocket's airship with Explosion, and with fighting Deoxys. Since then, it is back on Cinnabar Island, training with Blaine.
Mewtwo has an Anime/Manga exclusive form named Armored Mewtwo in the anime, and M2 Bind in the Manga. In the anime the form increases all of Mewtwo's Psychic abilities, and in the Manga it allows Giovanni to control it.
Mewtwo also has an Awakened form due to appear in the Pokemon X and Y games, and making its anime debut in the movie: Extremespeed Genesect: Mewtwo's Awakening. The form drastically changes Mewtwo's appearance, removing its tail, giving it a long head, and changing it's eye colour, feet and hands.
Mewtwo's name comes from the word 'Mew', which is like a cat meowing, and 'two' because it's a clone of Mew. It is based on a DNA scientific cat.
Although it is a clone of Mew, it comes before Mew in the Pokédex possibly because Mew is a Mythical Pokémon and Mythicals always come last in the PokéDex.