This information was compiled after carefully studying the events of the game, along with some additional research done through the Web.
(SPOILER ALERT!!! Don't read past the marker if you want to figure things out on your own.)
As you may already know from the reports, the cycles began in World A, when Chaos, a sentient Manikin created by Onrac researchers with help from Cid, a brilliant Lufenian scientist responsible for the invention of airships in their world, was taken away to be used as a weapon of war from his guardians, Cid and his wife. To force Chaos into obeying, they created another sentient Manikin resembling Cid's wife to take charge of him. This Manikin would be known later on as Cosmos.
Through the wars he was put through, Chaos eventually becomes hateful. When Cid and his wife attempt to rescue Chaos from Onrac, Cid's wife is killed, causing Chaos to finally snap, his power causing an opening in the Interdimensional Rift that sucks him, Cid, and Cosmos into World B.
Stuck there with no idea how to get home, the three are offered a way out by Shinryu, a dragon with a lust for power, who lived in the Rift and happened to observe their arrival. The dragon had sensed Chaos' immense powers and wanted them for his own. Shinryu promised them a way to get back home if they made a pact with him.
The pact made, Chaos and Cosmos gained the ability to summon warriors from other worlds to battle each other until one side loses. Cid, being human unlike the other two and thus may not live through all the cycles, was turned immortal to act as an observer, at the cost of losing his physical body. The powers of all the fallen warriors would in turn be absorbed by Shinryu as payment for the pact, while Shinryu would imbue some of this same power to Chaos. The cycle would then repeat, with the warriors' memories of the last cycle wiped clean. This process would gradually give Chaos the power to finally open a portal back to World A.
At first, Cid was content in observing the cycles, whilst continuing his research on Manikins, trying to create a perfect being by transplanting his own memories into it, thus giving it sentience and will. In the end, the best he could create was one who was a perfect likeness of him, but had no will of its own, which Cid had attributed to a failed transfer of his memories. He left this Manikin amidst the conflict to see what would become of it. This Manikin would, in the future, be known simply as the Warrior of Light. The rest of his creations, imperfect Manikins that were crystalline, held no human memories, and desired nothing but conflict, were discarded into the Rift.
Chaos, reluctant to fight with someone he pictures as his mother, allowed his warriors freedom to do as they pleased, thus causing some of them to have their own hidden agendas. Some of them wished to rule World B, some wished to destroy it. As the cycles continued, he slowly lost his memories of his time at World A, but still remembered his relationship with Cosmos through vague dreams.
Cosmos, on the other hand, willingly complied with the cycles at first, but slowly despaired as she observed her warriors fighting for their hopes and dreams despite not having any memories of how many cycles they've been fighting, not knowing how much they've gone through.
By the time of Cycle 012, Cid had realized the same thing as Cosmos. Breaking his pact with Shinryu, he stopped being a mere observer and sided with Cosmos, helping her by formulating a plan to end the conflict using crystals formed with both Cosmos' and the warriors' powers.
The plan was supposed to proceed during that cycle, but the imperfect Manikins complicated matters, released from the Rift by one of Chaos' warriors, Exdeath, who wished to sink the world into the Void. In the end, the Warrior and another from Cosmos' side, Kain Highwind, hid the other warriors from the Manikins. 5 other warriors of Cosmos joined Kain after that, abandoning the conflict to close the tear in the Rift the Manikins were coming from, thus greatly limiting the Manikins' numbers, at the cost of their lives. As a result of their deaths outside of the conditions Shinryu set for all in the conflict, they lost the chance to be revived in future cycles.
With the Warrior the only one remaining unhidden, defending Cosmos from the remaining Manikins who by now had reached Cosmos' fortress, Cosmos sacrificed herself to destroy the Manikins, saving the Warrior in the process, and ending Cycle 012.
Angered by Cid's betrayal, Shinryu revives Cosmos and the fallen warriors on both sides (except the ones who died closing the Rift), and sides with Chaos in Cycle 013 to thwart Cid's plan. Cosmos, having been revived with no memories of her plan to end the cycles, was about ready to resign herself into continuing until a traitor to Chaos' side, Golbez, reminds Cosmos of her plan, as the warlock also desired an end to the cycles, both for himself and also his brother, who happened to be a warrior of Cosmos.
The combination of Cosmos sacrificing her life once more, this time to imbue power to her warriors through their crystals, and the resolve of her warriors, led to Chaos finally being defeated, ending the cycles at last and transporting the warriors back to World A, and later on to their respective worlds. The Warrior, having no world of his own, remained at World A, making it his home.
Further angered by this loss, as ending the conflict would mean no more power from fallen warriors for it to consume, Shinryu traps Cid in a nightmarish realm resembling World B, where events happened differently and Chaos survived Cycle 013, but kept on suffering losses on each subsequent cycle. By Cycle 018 in this realm, Chaos had been driven insane by the losses and entered the conflict himself, destroying both Cosmos' warriors and his own, turning into what is now known as Feral Chaos.
By Cycle 020, Cosmos had already weakened to the point that she could no longer summon warriors to her aid. With Chaos on the way to destroy her, Cid sealed Cosmos in World B's Cavern of Earth. Being denied his target, Chaos instead turned on Shinryu, challenging the dragon. Cid, in turn, sealed off the entire northern continent of World B to restrict the conflict to that part of the world. He then sealed himself as well.
Eventually, the fighting died down, and Chaos himself went dormant.
Some time after that, 5 warriors were summoned by an unknown force into World B. By this time, only Moogles and Manikins roamed the land. The warriors eventually found Cid, now residing in the body of a Moogle with almost no memories of events prior. Using what little he remembered, he led the warriors to the northern continent, his memories slowly returning as the journey continued.
By the time they arrived at World B's version of Onrac, where Chaos was, Cid had finally recalled everything, and asked the warriors to destroy Chaos and finally free him from this realm. The warriors complied and, after an intense battle, finally managed to defeat Chaos.
As Chaos perished, he regained his senses, and thanked everyone for freeing his "father" from the nightmare. The warriors returned to their own worlds, and Shinryu, defeated, flew back into the Rift, thus ending the cycles for this nightmarish alternate reality.
As the alternate World B began to crumble, Cid asked Cosmos, now unsealed, if she wanted to come back with him to World A. The goddess of Harmony declined, stating that she would prefer to stay in that realm until it ended. Cid, respecting her decision, left her and returned home.