The Lethe is one of the five rivers of The Underworld, which is ruled by Hades. When souls or as the Ancient Greeks called them, shades, were in the underworld, they lived in a sort of 'half consciousness', continuing to labor and emote as they did on earth while wandering the Fields of Asphodel awaiting judgement by Hades and his wife Persephone, but without the vim and vigor of, say, a living human.
It is
said that river was created by Hades as a gift to his citizens, since drinking from it swept away all of their memories of life on earth.
Not only would they forget their wants, troubles and pain, but they would also forget their loved ones and the joys they felt on earth in order to spare them the pain of missing them. Some also believe that the shades who were destined to return to earth in different forms, via reincarnation were instructed to drink a certain amount of the water so they could return as a 'clean slates', free of knowledge or pre-judgement.