Short answers, pulled from the story.
The Erinyes are chthonic goddesses of vengeance in ancient Greek religion. They live in Erebus and are older than the Olympian deities. Their role was to punish mortals who violated sacred obligations, including false oaths, dishonor toward parents, and abuse of guests or suppliants.
The three Erinyes named by Virgil are Alecto, meaning endless anger; Megaera, meaning jealous rage; and Tisiphone, meaning vengeful destruction. Virgil likely drew on an Alexandrian source when he fixed their number at three and gave them individual names in the Aeneid.
The Furies are the Roman counterparts of the Greek Erinyes. The Roman writer Maurus Servius Honoratus, writing around 400 AD, recorded that the same beings were called Eumenides in hell, Furiae on Earth, and Dirae in heaven.
According to Hesiod's Theogony, the Erinyes were born from drops of blood that fell on the Earth when the Titan Cronus castrated his father Uranus. They emerged alongside the Giants and the Meliae from the blood that struck the ground.
In the Oresteia, the Erinyes pursue Orestes after he kills his mother Clytemnestra to avenge his father Agamemnon. They serve as his accusers at a trial in Athens presided over by Athena. When the jury splits evenly and Athena votes for acquittal, Athena persuades the Erinyes to accept a new role as protectors of justice and of Athens.
The name Eumenides, meaning the Gracious Ones, was a euphemism used to avoid attracting the Erinyes' attention by naming them directly. Euripides was the first writer to formally equate the two names in his Orestes. The practice followed the same logic as calling Hades by the name Pluto, meaning the Rich One.