Mewtwo (ミュウツー, Myūtsū) is a fictional creature from the popular and expansive Pokémon franchise. It is introduced as the "final Pokémon" in the very first game in the franchise. Mewtwo enjoys a good degree of fame and popularity in the Pokémon fan community for its significance as the original "Uber Legendary", and it is included in Super Smash Bros. Melee as a playable character. It is one of the five characters to not return as playable in Super Smash Bros. Brawl however.
Mewtwo is Pokémon #150 in the Generation I Pokédex, the final Pokémon in the list not counting the generally-unobtainable Mew which is #151 in the Pokédex. It is a clone of Mew that has been altered by unscrupulous scientists in order to increase its power and ruthlessness in battle. As a result, it is visually similar to Mew: a pale purple, vaguely feline humanoid with strong psychic powers. Unlike Mew, Mewtwo has the height and mass of an adult human, and thinks of nothing but fighting and defeating its opponents.
In-game, Mewtwo was designed to be the "final" Pokémon of the Generation I games, having a immense Special stat and a very high speed stat, the highest base stat total with a total of 590, and the (at the time clearly overpowered) Psychic type. As a result Mewtwo was an overcentralizing force that unbalanced the metagame until the next generation's Special split into Special Attack and Special Defense, and altered type interactions evened the playing field. In Generation VI, Mewtwo gained access to new transformations called Mega Evolutions. Like Charizard and unlike most Pokémon who can also Mega Evolve, it has two Mega Evolve forms if it is holding a Mewtwonite X or a Mewtwonite Y as its item: the larger and bulkier Mega Mewtwo X, and the small, light, and floating Mega Mewtwo Y respectively.
In the Pokémon anime and movie series, Mewtwo is a (at the time of the first movie) one-of-a-kind Pokémon that develops a hatred for humans and the concept of subserviant Pokémon, attempting to prove the superiority of clones over originals. It later has a change of heart and becomes a sort of anti-hero, mostly keeping to itself and trying to find its purpose.
Mewtwo does not appear in the original Smash in any sort of capacity. However, according to a page on the Japanese Super Smash Bros. site, Mewtwo was planned to be included as a playable character, but was cut; Bowser and King Dedede are also mentioned as having been planned but cut, though no other information is available. 
In Super Smash Bros. Melee
As a playable character
Mewtwo made its Smash-series debut as an unlockable character in Melee. It is much more tedious to unlock than most other secret characters in Melee, however; it can be unlocked after either playing 20 cumulative hours of Vs. Mode matches (5 hours for four players), or playing 700 Vs. Mode matches.
Mewtwo's playstyle is very floaty and includes a decent variety of combos, strong throws, a very good wavedash, above-average dodges, a strong projectile, and one of the best recoveries. On the other hand it has a bad combination of large size and light weight, which results in it being an easy target to hit and combo while being especially easy to KO, giving Mewtwo some of the worst survivability in Melee despite its recovery and defensive prowess. In addition to this crippling problem it has general awkwardness in its moveset and movement, most of its moves being not very useful or outright near useless attacks, lacking solid KO options, and having slow movement speed. As a result, Mewtwo has infamously sat in the bottom tier of Melee's tier list for most of the game's lifespan, being among the game's most unpopular characters and considered the worst character at several points in the game's lifespan, having only recently climbed its way out of the bottom tier to its current position of 21st.
Mewtwo, as a playable character, has three trophies - a main trophy acquired by defeating the Classic mode with Mewtwo on any difficulty, and the two Smash trophies are by beating the Adventure and All-Star modes, respectively. Its Classic mode trophy reads:
Despite not being a clone at all like the other cut characters and having one of the most unique character designs, Mewtwo did not return in Brawl as a playable character, instead just appearing as a trophy. This resulted in a significant amount of fan outcry, despite Mewtwo's lack of popularity as a playable character in Melee, as the majority of players saw it as an unjustified cut. Unfinished data of Mewtwo has been found on Brawl's disc however.
Mewtwo appears in Brawl as a collectible trophy, obtained by completing All-Star Mode on Intense difficulty. The trophy describes Mewtwo's appearance in the Pokémon games, and makes no reference whatsoever to its appearance in Melee:
While Mewtwo does not appear as a playable character in Brawl, data of the character has been uncovered through hacking, as ef_mewtwo.pac. The exact reason why the data is included is unknown, though it is commonly believed that Mewtwo was planned to be included as a playable character, but time constraints prevented Mewtwo from being finished and resulted in it being cut.
Of particular note is that the data for Mewtwo has its own unused fanfare, as "snd_bgm_Z38_MYU2". Many other claims have been made about the progress of Mewtwo's data, such as an included character model and even parts of its voice, though these claims remain unverified. To date Mewtwo has had the most claimed unused content of any scrapped character.
In Super Smash Bros. 4
During the Pokémon X and Y Developer Roundtable at E3 2013, a question was asked of the developers whether they would like to see Mega Mewtwo appear in the next Smash Bros.. While the developers were unsure what to answer and replied with a generic lack of comment, Sakurai happened to be in the audience, and was claimed to exclaim "We are thinking about it.". While neither Mewtwo nor Mega Mewtwo have yet been confirmed or denied for Smash 4, this statement, combined with the aforementioned fan outcry over its cut from Brawl, and Mewtwo's recent renewed prominence in the latest generation of the Pokémon series, brings many fans to believe a comeback is more likely than not.