Okay, so we know it sounds a little weird to have a play where its characters are symbols. But really, this happens all over the place. If you've ever read, say, Lord of the Flies in your English class, you've probably had your teacher drum into you that the character Simon is a "Christ-figure," which is just a fancy way of saying that he "symbolizes Christ."
Or, you've probably seen in a cartoon or on TV, where a character will be debating between two courses of action, and a devil and an angel appear on each shoulder. In this case, the writer of the cartoon has decided to symbolize two ideas (the good and bad courses of action) by creating two characters (the angel and the devil) that represent them. So, actually, this sort of thing is common.
What about the Furies? Well, they are goddesses of revenge. The way they are described makes them sound totally freaky and disgusting:
(Prophetess): "In front of this man an amazing band is asleep, of women, sitting on the chairs—no, I do not mean women, but Gorgons […] these however have no wings to be seen; and they are black, utterly revolting in their manner, snoring out a breath which is unapproachable […] (46-59)
This nasty bit of description—the Furies clearly have to up their oral hygiene game—is Aeschylus's way of saying (symbolically) that revenge itself is freaky and disgusting. They are "black," a color which had the same symbolic resonance in Ancient Greece as it does in much of the Western world today—you wear black at funerals, a bad guy is said to be "black-hearted" and nasty or violent thoughts are described as "dark" or "black."
Also, it's worth noting that they have no wings. This underlines the fact that they're immobile (and, as we see with the binding enchantment, capable of immobilizing their enemies). Don't think for an instant that this is unintentional—the spirit of revenge, much like The Furies, renders people immobile with single-minded obsession.
When the Furies turn nice at the end of the play, that's Aeschylus's way of symbolically saying that the rule of law, as expressed through law-courts, takes away the nastiness associated with the old customs of revenge.