The game's main plot is the same as that of Undertale, with the war between Humans and Monsters, and the latter being sealed inside the Underground via a magic barrier. However, it follows the journey of the seventh human that falls into the Underground. This human is a young child named Clover.
An ambiguous amount of years after the war, multiple humans have been rumored to disappear after heading toward Mount Ebott. With a Missing Poster in hand, Clover heads toward the mountain to investigate, and deliberately jumps into the Underground. There, they team up with Flowey and explore the Underground, with Flowey offering to Save for Clover.
Because Flowey is present in the game, it is possible that Undertale Yellow takes place just after the experiments that led to the creation of the Amalgamates. This may or may not imply that Alphys will make a reappearance.