How were the Erinyes born? They seem like mystical creatures that are there to do Hades' bidding, but how were they actually born? Are they just other monsters magically born?
According to Hesiod, the Erinyes were born of Gaia (Earth) when she received the blood of Ouranos (Sky):
And Heaven came, bringing on night and longing for love, and he lay about Earth spreading himself full upon her. Then the son from his ambush stretched forth his left hand and in his right took the great long sickle with jagged teeth, and swiftly lopped off his own father's members and cast them away to fall behind him. And not vainly did they fall from his hand; for all the bloody drops that gushed forth Earth received, and as the seasons moved round she bore the strong Erinyes and the great Giants with gleaming armour, holding long spears in their hands and the Nymphs whom they call Meliae all over the boundless earth.
Source: Hesiod, Theogony
Their most popular origin story is the gory version in Hesiod's Theogony, in which they are born from Gaia, the Earth, when she is impregnated by the blood shed by Ouranos [Uranus, the "Sky"] while he is being castrated by his own son the Titan Kronos.
Aeschylus, however, in his play Eumenides, whose title is a euphemistic epithet of the Erinnyes, says that they were born the same way most of the vices and evils which afflict the human race were born, when Nyx, "Night," without the assistance of male seed, engendered Death, Old Age, Pain, Deceit, Envy, Injustice and various other such daimones. Among these were the Erinnyes. This is also Lycophron's version of their origin while the Roman poets Virgil and Ovid have the Furiæ as the daughters of Nox (the Roman Nyx). This version makes these goddesses older than (or at least as old as) the Titans.
Writing in Latin, Hyginus lists the Furiæ together with the Titans and Centimani ("Hundred-Handers") as though they are fairly natural offspring of Cælus and Terra [the Roman equivalents of Ouranos and Gaia respectively], making no mention of the violent account we are offered by Hesiod. Hyginus, however, does have a rather ambiguous writing style, and so Robert Graves, in his book The Greek Myths, interprets that passage of Hyginus' Preface as meaning that the Furiæ are the offspring of Terra by Æther (the bright, clear air which exists above the clouds and which only the gods can breathe safely; the ancient conception of ozone, so it would seem).
According to two Orphic hymns, the Erinnyes are the daughters of Haides [Hades] and Persephone.
The Roman writer Statius, in the Thebaid, says that the father of the Furiæ is Erebus, who would usually be "Darkness," the husband of Nox, but Aaron Atsma of The Theoi Project, for some reason, interprets this as a reference to Dis (the Roman Haides).
Another Roman, Valerius Flaccus, in Book 1 of his epic Argonautica, makes a passing reference to Pœna, "Retribution" [from whom we get the English word subpoena], being the mother of the Furiæ.
Combining the last two Roman sources would thus give us "Erebus" and Pœna as the parents of these goddesses.
While the Erinnyes are certainly portrayed as cruel and unrelenting, they are, however, never presented as irrational monsters nor as especially mystical in an inexplicable way. They are in fact the personification of an aspect of culture that was (and actually still is) perceived as necessary in order for society to continue functioning in the face of offences which might be committed against authority figures, basic family structures, and whatever is considered to be the natural order of things in general. They represent the dark or violent side of balancing things out by righting wrongs through, e.g., blood vengeance for the murder of a close relative, a concept which is commonplace in many other cultures around the world (see the rules about the Avenger of Blood in the Mosaic Law code [Numbers 25, Deuteronomy 19 & Joshua 20], for instance).