Unlike previous Smash Bros. titles, Endeavor's Adventure Mode is split into multiple Stories, some of which (such as the classic Subspace Emissary) are included as DLC.
New Story ~ Darkest Hour
The primary story mode of Endeavor and the longest of any non-DLC Story, this mode features every character and most bosses in the game at some point or another, making it very similar to Subspace Emissary from Brawl. However, Darkest Hour is set in a completely different world which follows different rules - for one, defeated fighters do not turn into trophies under normal circumstances. This is the only Story that will unlock characters when they join one's party that isn't a DLC Story - all other Stories included in the vanilla game have no effect on the character roster in Vs. Mode. An overview of this Story can be found here.
Old Story ~ Subspace Emissary (DLC)
A reproduction of the Adventure Mode from Brawl, this Story follows the cast from Brawl (excluding Toon Link, Wolf, and Jigglypuff) on their journey to defeat the armies of Subspace, led by the villainous likes of the Ancient Minister, Ganondorf, Bowser, Wario, and the mysterious Master Hand...
UnderGround ~ The Reaper's Reprise (DLC)
A Story based around the DLC from The World Ends With You; the hero this time, Neku Sakuraba, finds himself dragged into another Reaper's Game. But this time, the rules are a bit different... as are the Players! Ally yourself with Nintendo's all-stars in a seven-round mission completion quest as you go up against the Noise and, at the end of it all, the Game Master himself.