Discordia is the Roman version of the Greek goddess Eris; there are no actual distinctive differences between one character or the other. As a matter of fact, the name “Discord” or Discordia, is only the Latin translation of the Greek name or word Eris. In Discordianism however, the decidedly nonsensical religion started in the 1950’s, the primary goddess is of course, Eris, but she’s actually referred to in the Principia Discordia as “Eris Discordia”; rather than use one or the other, they’ve incorporated both names. Other than in the “religious” texts of Discordianism, and other pop culture references, Discordia doesn’t really appear in any strictly Roman mythology.
The Greek goddess Eris on the other hand, is a different story. According to different works by Hesiod, Eris was duplicated; instead two goddesses. One was believed to be a daughter of Zeus and Hera, an equate to Enyo, the sister of Ares, although Hesiod does not make this clear. This other is a daughter of Nyx, and Cronos. The first daughter, “Strife”, is a deity representing the cruel and chaotic ways of war, while the second, also called Eris/Strife, represents the strife of men to work hard. Despite the face that the daughter is Nyx and Cronos is spoken of by Hesiod to be more helpful to mankind, later on he gives her a less polite description. According to Hesiod, the Eris/Strife that was born of Nyx and Cronos had several children:
"But abhorred Eris (‘Strife’) bare painful Ponos (‘Toil/Labor’), Lethe (‘Forgetfulness’) and Limos (‘Famine’) and tearful Algea (Pains/Sorrows), Hysminai (‘Fightings/Combats’) also, Makhai (‘Battles’), Phonoi (‘Murders/Slaughterings’), Androctasiai (‘Manslaughters’), Neikea (‘Quarrels’), Pseudea (‘Lies/Falsehoods’), Amphillogiai (‘Disputes’), Dysnomia (‘Lawlessness’) and Ate (‘Ruin/Folly’), all of one nature, and Horkos (‘Oath’) who most troubles men upon earth when anyone willfully swears a false oath."
Eris is also known in another story to have caused sufficient chaotic and confusing problems for people. The married lovers Aedon and Polytechnos claimed to love each other more than Hera and Zeus loved one another. This caused Hera to become quite annoyed, so she sent Eris to cause trouble for the couple. Eris came to Polytechnos and Aedon while they were at separate tasks; Aedon was weaving embroidery and Polytechnos was making a chair. Eris said that whomever should finish their task first must receive a female slave from the other. Aedon won the contest with Hera’s help, and Polytechnos was furious. In his anger, he took Aedon’s sister from her father’s house, raped her, and dressed her in a slave’s disguise before delivering her to his wife as the prize. Later, Aedon discovered her husband Polytechnos’s deed, and the two sisters conspired against him. For revenge, the two sisters murdered Itys, Polytechnos’s son, cooked him, and fed him to the unaware Polytechnos.
Afterward, Aedon and her sister fled back to her father’s house. When Polytechnos chased his wife there. Aedon’s father had him tied up, smeared with honey, and hung naked outside to be tortured by insects. When Aedon saw how cruelly her husband was being tortured, she lamented his suffering. Her relatives were quite irritated, and began conspiring to kill her for her weakness. Zeus took pity on the quarreling and confused family, by changing them all into birds. Polytechnos became a pelican, Aedon a nightingale, her father a sea-eagle, her brother a whoop, and her sister a swallow. According to mythology the nightingale’s singing is a lament for having killed her own son in revenge against her husband.