The Sound stone is a Key item passed to Ness by Buzz Buzz during the opening events of EarthBound. The Sound stone acts as a kind of recording device, and is able to contain the power of the melodies that Ness and his companions hear at each of the eight "Your Sanctuary" locations. Until it is completed, Ness can use the Sound stone to hear the melody in whatever state it is in at the time.
When the last melody is absorbed by the Sound stone, Ness appears to fall unconscious, yet at this time he is actually venturing into Magicant, a place created purely from Ness's mind and memories.
Upon arriving at the center of Magicant (and defeating the evil that resides there), Ness regains consciousness, and the Sound Stone shatters, as its purpose is fulfilled. This also results in Ness absorbing the power of the eight Sanctuaries, as well as his own inner potential, granting him the power needed to complete his journey.
It should be noted that if Ness's inventory is full when he needs to receive the Sound stone, Buzz Buzz gives the stone to Ness's sister, Tracy, freeing up an inventory spot for the rest of the game, this is quite useful because you are considerably limited in inventory space until you recruit more characters. The Sound stone still works normally, recording the melodies of the "Your Sanctuary" locations, even though Tracy is in possession of the stone. If Ness wishes to obtain the Sound Stone back, he can talk to Tracy in their house. It is not possible to get the item back through the Escargo Express.
The Sound Stone is also instrumental to travel to other places on the map with the "Walk Through Walls" code