In Greek mythology, which educated people of Eliot's generation knew well, the Eumenides or Furies, were goddesses of revenge. They were often conceived of as ghosts of murdered people, sent to torment the guilty.
In Eliot's play, Harry has murdered his wife by pushing her overboard on a ship. He gets away with it because it is deemed a suicide, although Downing, his servant, will frankly state that he doesn't thinks she took her own life.
Harry, who Eliot likened to Hamlet, is tormented by sightings of the Eumenides after committing the murder. He believes these Furies are in pursuit of him. At Wishwood, the ancestral family home which is the site of the family reunion, Mary, Agatha, and Downing see the Eumenides too. These three characters can discern the Furies because they are people not asleep or living in the past. They can therefore perceive spiritual reality, unlike Harry's mother, Amy.
The Eumenides are important because they represent the spiritual aspect of life that walks with us, though not everyone is aware of it. They prod Harry in the right spiritual/moral direction, helping him to break away from Wishwood and the past.
Harry is freed of the curse of Eumenides when he realizes that his father, like him, had plotted to kill his wife. His father, however, had not gone through with it because his lover and wife's sister, Agatha, talked him out of it. Amy was pregnant with Harry, and Agatha couldn't bear the idea of the baby's death.