Taking place after Eric skewered Ursula in the original film, Ariel and Eric are planning to wed, but the sea-witch Ursula (somehow resurrected) has taken control of the ocean. So Ariel becomes a mermaid once more and sets off to rescue the sea.
The game takes place from a side view and Ariel (swimming most of the time, but hopping around on the land occasionally) can shoot bubbles to trap her foes and can then throw them at each other. She can also dig through sand to find treasure and pick up sea shells to break chests open with.
Although this game is not canon, this exact situation where Ariel had to be turned back into a mermaid would be duplicated in The Little Mermaid 2: Return to the Sea where she dives off a much larger boat (Eric's lead fleet ship) rather than his rowboat in the NES game.
Source: Wikipedia, "Ariel (The Little Mermaid)," available under the CC-BY-SA License.