In Super Smash Bros. Brawl, what is the name of the story mode?

In Super Smash Bros. Brawl, the story mode is called the Subspace Emissary.

Officially titled “The Subspace Emissary,” this Adventure mode gives Brawl an interconnected single-player storyline. It combines side-scrolling platform levels, boss fights, cutscenes, exploration, and a large cast of playable Nintendo characters. Players begin with a limited group, while other fighters join the campaign as the story progresses.

The plot follows the heroes as they oppose the Subspace Army, whose attacks use Shadow Bugs and Subspace Bombs to pull parts of the world into an alternate dimension. The mode can also be played cooperatively, making it more substantial than the traditional Classic or All-Star modes. A common mix-up is calling it simply “Adventure Mode”: Adventure is the broader mode type, while The Subspace Emissary is Brawl's specific story campaign. Tabuu is the campaign's final antagonist.