Who is He?
The main antagonist of the "Subspace Emissary" adventure mode of Super Smash Bros. Brawl, and the secondary antagonist of Super Smash Bros. Ultimate (albeit as a spirit fighter). He is the entity who resides in the Subspace that wishes to tear the Super Smash Bros. world into pieces and drag it into his.
What has he done?
Although Tabuu does not appear until late in the story mode, Tabuu's unseen presence is apparent while he is controlling Master Hand behind the scenes. After Ganondorf turns Bowser into a trophy, he discovers that Master Hand is being controlled by Tabuu. Tabuu turns Ganondorf into a trophy and easily defeats Master Hand.
Once Mario, Link, Kirby, Pit, Yoshi, Peach, Zelda, the Ice Climbers, Lucas, Captain Falcon, Captain Olimar, Donkey Kong, Diddy Kong, Samus, Fox, Falco, Meta Knight, Lucario, Mr. Game & Watch, Pikachu, Pokemon Trainer, Ike, Marth, R.O.B., and Snake all confront him, he turns every one of them into trophies easily, seemingly leaving no one to stand against him. However, King Dedede's medallions turn Kirby, Ness, and Luigi back to normal, and they turn him back in return (since he was turned into a trophy by Bowser). They track down everyone else in Subspace and restore them. Bowser, Ganondorf, and Wario join their team and eventually they reach Tabuu again. He is about to use his wings to turn them into trophies again when suddenly Sonic appears and destroys Tabuu's wings in four seconds flat. He joins their team, and together they fight against Tabuu. Finally, after much effort, Tabuu is finally defeated, and upon his defeat, he shrieks and disintegrates as light energy erupts from him and the Subspace fades into nonexistence.
Tabuu returns in the "World of Light" adventure mode in Ultimate as a legendary class neutral type primary spirit (the most powerful spirit type), having presumably been destroyed by Galeem and stolen by Dharkon, forced into doing the latter's bidding, implying they are both more powerful than even he is. He is one of the spirits the fighters must defeat in order to advance through the rest of the game. Tabuu is found in the Mysterious Dimension possessing a metal puppet fighter of Bayonetta and must have his stamina depleted in order to obtain him. Periodically, the player's stamina will drop suddenly as a reference to Tabuu's off-waves. After defeating him, the fighters will engage in a boss fight against Marx.
It is unknown if Tabuu had regenerated only to be killed in one shot by Galeem or if he still possessed no physical form at the time. It is also unknown whether Ultimate and Brawl are in the same continuity, or whether Tabuu's soul was claimed by Galeem upon his defeat in Brawl.