Introduction - Mythological Genealogy
Making an overview of Hesiods "Theogony", everything came to existence thru Chaos, which had 4 offsprings: Gea, Eros, Erebus and Nyx.
Gea birthed the Titans (which later on birthed the Olympians): Olympus, Tartarus, Uranus, Oreos, Pontos
Eros had no offsprings.
Nyx gave birth to: Moros, Ker, Thanatos, the Hesperides, Hipnos, the Moireae, Sleep, the Keres, Momo, Ezis, Nemesis, Apathe, Filotas, Gera and Eris.
Nyx had offspring with Erebus: Ether and Hemero.
Now that we have a general perspective, we can focus on the myths that involve Eris: the battle of Typhon agaisnt Zeus and the Judgement of Paris.
The battle of Typhon and Zeus
When Zeus defeated all of the Titans, Gea and Tartarus bore Typhon (and Echidna), and summoned him to defeat Zeus, for Gea did not wish her Titan offsprings to live in imprisonment; Zeus, with his newly gained power, was not eager to let go of it, thus, battled Typhon.
Typhon was escorted by Eris to battle while Zeus was by Nike; Typhon initially overpowered Zeus, but the struggle turned in favor of Zeus at the use of his thunderbolts, and finally imprisoned Typhon alongside the majority of the Titans, under Tartarus.
I think it is important to hold off the analysis until both myths have been exposed.
The Judgement of Paris
The "judgement of Paris" is but a part of a bigger event, which was the wedding of Tethys and Peleus.
Having the Olympians banished the majority of the Titans to Tartarus, they shared the power and control over everything within the Universe (except the primal deities); now the only obstacle to keeping their power as status quo, was the fact that the offspring of Tethys was phropecized to be "stronger than his father". Zeus and Poseidon (Leader and 2nd in command), really wished to "posess" Tethys, but decided that neither one of them should have her and should marry a mortal, so there would be no threats to the Olympian hegemony. That mortal was Peleus.
The wedding of Tethys and Peleus was an arraigned marriage, of which Tethys was unhappy about, but had no say in the matter. All the gods were invited, except Eris, the matron goddess of strife and discord, so she wouldnt summon the discontent of Tethys.
Eris, being uninvited to the party, decided that if at least she was not gonna be present to give the bride her best wishes, she should at least drop off a gift, and so she did. "For the most beautiful/fairest one", an apple made of pure gold.
But Tethys never received her gift from Eris, because the 3 most "important" goddesses (Hera, Aphrodite, Athena) decided that, if theres is anyone that is "beautiful/fair", it was them.
Paris was summoned to decide, for him being the least corrupt of the humans, who was the one deserving of said apple. And lo and behold, he was offered kingdoms, military skill, but he chose the most beautiful woman on the planet.
And thus was the first war amogst men.
Making a quick analysis of this; Zeus is the personification of a ruler that has overthrown the previous ruler, his own father Cronos, which in turn had overthrown his own father, Uranus.
In order to break the cycle of violent wars and overthrown rulers, his method was consensus:
-After the Titan-Olympian war, he split the dominion in three parts with his most powerful brothers and himself: Earth, Ocean and Underworld (Zeus, Poseidon and Hades respectively).
-He married off Tethys to a mortal.
His victory over Typhon was symbolically, the triumph of humanity over nature; the children of Gea and Uranus prevailed, while the children of Gea with Tartarus, such as Typhon and Echidna were respectively banished to Tartarus, or the lesser ones that didnt pose a real threat to his rule as mere challenges for the human heroes. Thus order was established, and as it had no challenge, grew unchecked.
The final nail to the coffin of disorder/change was the marriage of Tethys (being an earlier goddess that got integrated into Greek mythology). But although it managed to keep order amongst the Gods, it only provoked disorder among mortals. Thanks Zeus!