Mewtwo is the Legendary Genetic Pokémon, one of the most powerful and notoriously man-made Pokémon in existence.
The Original Mewtwo
Unlike most Legendary Pokémon, Mewtwo had gone off the radar long before Team Rocket took over. At one point during their reign, Mewtwo temporarily left his colony of clone Pokémon. In his absence, a Team Rocket operative managed to locate the colony and, using a machine that amplified the power of his Pokémon, killed the majority of the clones. Mewtwo returned, and after being informed of what happened by the remaining clones, fell into a rage, hunting down the Rocket agent. It is unknown what happened afterwards, though it is implied that Mewtwo destroyed both the machine and the agent responsible for the massacre.
Mewtwo came into contact with Ash and Misty Ketchum, who had been driven into hiding by Team Rocket. Mewtwo helped the pair go into hiding, occupying a cave along with a group of Clefairy soon after. Mewtwo is known to have stayed in contact with Misty, though it is not known why the same wasn't done with Ash.
When Team Liberty members Kyle Eston, Darren Ketchum and Gecynde Wynn arrived in Cerulean City so as to search for Misty, Mewtwo sensed their approach, and used a psychic link between himself and Gecynde's Cleffa to bring them to him. Using a group of Clefairy that occupied his cave, he evolved Gecynde's Cleffa via Moon Stone. The Pokémon then brought the group before Mewtwo, who at first simply appeared as a cloud of darkness. Dropping the veil, he engaged in conversation with the agents, relating the fate of his clones to them. Using this as motivation, Mewtwo agreed to ally with Team Liberty, before teleporting the group to Misty. After managing the convince the ex-Gym Leader to ally with Team Liberty, Mewtwo teleported the group to Four Island, where they helped defend Team Liberty's main headquarters during Team Rocket's invasion.
After the creation of the second Mewtwo, Team Liberty's Mewtwo began training with Liberty Executive Jayron Aaron so as to learn the dimension-manipulation abilities of his counterpart, which Jayron had observed during his meeting with Dara and the second Mewtwo in Unova. The two briefly managed to sense the disturbances caused by the other Mewtwo's attacks in Azalea, though could not track the two down as they had hoped. At some point during their training, the pair were attacked by a group of Team Energon Agents, though with the help of Team Liberty Executive James Arthur, they managed to fight them off.
The Second Mewtwo
A female Mewtwo is known to have been created by another group of scientists sometime before Team Rocket's rule, eventually escaping and coming into conflict with a group of Genesect in Unova. Its status after this point is unknown, as it has not been seen since.
The Third Mewtwo
As part of his plan to boot Team Liberty and Team Rocket out of Johto, Hoenn, and Sinnoh, John Ford commissioned the creation of another Mewtwo through the bounty hunting organization headed by Kevin Sparks. Upon its creation, Sparks began testing its powers, though took heed not to overdo it, only ensuring that it was capable of what Ford had asked. After confirming this, Sparks gave it to Dara Trina for delivery, trusting that her strong bonds with Pokémon would ensure that she would take good care of it. As a semi-final test, Dara decided to use Mewtwo to help quell a Shadow Genesect, eventually succeeding with assistance from a number of Team Liberty and Team Rocket trainers.
Dara received orders to meet Quincy Adams in the ruins of Azalea Town in order to finish Mewtwo's testing and to complete the trade. There, Adams began to battle Mewtwo, with Dara providing directions to the inexperienced and newborn Pokémon. As the battle progressed, Dara and Mewtwo began to overuse Mewtwo's dimensional manipulation abilities to such a point that they threatened the already-fragile fabric of reality around Azalea Town, despite warnings from Adams. The resulting damage summoned Nine to the location, throwing the Ancient Darkrai into a rage in which he attacked and forcibly immobilized Mewtwo, knocking it unconscious. After chiding Dara for misusing the Psychic-Type's powers, the Hojohsin Agents departed, taking Mewtwo with them.
Adams was entrusted with continuing to train Mewtwo by Ford, and thus they began, working to increase Mewtwo's proficiency with his abilities. During this training, Adams began to refer to Mewtwo as "Mewthree." After some time of working at this, Adams was ordered by Ford to assess a situation involving Pryce in a side-dimension; he advised that she take a number of Déoza Agents with her along with Mewthree, and to test out a dimension-crossing device created by Déoza. This process involved the hijacking of a train--the Pleasant Meadows Express, which connected Violet City and Azalea Town. The process was jeopardized by intervention from a group of Rocket Agents consisting of Jackson Sang, Jamie Arthur, Lucia Zyther, Lafoyae Belrose, Gavin Braner, and Ethan Worth, but ultimately succeeded.
After arriving in the dimension, Mewthree and the others set out, soon encountering Pryce in the guise of the Mask of Ice. Pryce spoke cryptically to them before setting a number of strange entities upon the Hojohsin group, killing and injuring a large number of agents and their Pokémon. Adams called for a retreat, prompting Mewthree to open a portal through which the others could escape, with himself remaining behind to fend off the attackers. The creatures tore him apart and departed, leaving him for dead.
The Rocket Agents witnessed the tail-end of the action, and thus were surprised when Mewthree began to regenerate, repairing his body. He briefly explained parts of his capabilities and why they had come to the dimension, though did not go into great detail. Together, the group formulated a plan to deal with the situation, sending Sang, Belrose, and Mewthree back to provide an exit for the trapped train passengers while the others confronted Pryce. The three managed to get the passengers to safety just as the others returned being pursued by Pryce's creatures, having managed to bring back some humanity to Pryce and reveal to him what he had become. After fighting the creatures off, the group escaped, and Mewthree returned to Ford's base.
Having recovered, Mewthree met with Ford's combined forces to discuss their next steps. While Pierce, Adams, and McKinley worked in the main dimension, Ford planned to check in on Pryce's dimension himself to see what had occurred there, and took Mewthree with him. Though Ford was later stated to have come back, Mewthree has not been seen again since then.
Powers and Abilities
Regardless of which individual is being spoken of, Mewtwo is an incredibly powerful Pokémon, even for one of Legendary status. It possesses extremely potent psychic abilities, which are strong enough to demolish buildings and manipulate weather patterns on a large scale.
The original Mewtwo has had over twenty years to learn and master its abilities, and as such is a force to be reckoned with. He is an incredibly skilled strategist, possessing a mastery over battlefield tactics to the point of hardly ever needing to utilize the full extent of his powers.
The second Mewtwo is fabled for being powerful enough to fend off anywhere between one to four Genesect at a given time, and as such is thought to possess great power. As she has not been seen in years, her present-day capabilities have not been confirmed.
The third Mewtwo was created with the purpose of manipulating dimensional fabric in the event that Nine vanished; as such he possesses extra capabilities not seen in the other Mewtwo. Along with managing to hold up against the impressive powers of Quincy Adams's Pokémon, he was able to defeat a Genesect in Unova, primarily through his ability to rip open dimensional holes. However, his powers are compromised by the fact that he is still a newly made being, and as such relies more on brute force than anything. Ford's Mewtwo's greatest strength, however, lies in his regenerative capabilities. Even after being torn apart in Pryce's dimension, he was capable of fully regenerating his body, though this used up a lot of power. This Mewtwo has been given the nickname "Mewthree" by Adams.