Organization XIII is a persistent antagonist group throughout the Kingdom Hearts series. The group has had many different members and forms over the years, but it's always been the driving force working against Sora and his Keyblade wielding allies.
In a plot-twist heavy ongoing franchise that's lasted nearly two decades, it can be a challenge to recall every iteration of the Organization and its motives. Here's some primer on the origins of Organization XIII and what they've meant to Kingdom Hearts.
Origins Of An Organization
Organization XIII begins to take form in Radiant Garden under the tutelage of Ansem the Wise. Ansem took apprentices and put his time toward finding ways to ensure peace and prosperity was always within reach for his people. This continued until he and his proteges came across an amnesiac calling himself Xehanort. After taking in the curious stranger, Ansem's goals shifted toward studying human hearts and the Darkness that resides within, hoping this might lead him to a breakthrough in understanding Darkness and provide insight into Xehanort's malady.
He'd construct a secret lab in the basement of his castle, and there he and his students began their ethically questionable experiments on human hearts. Soon after recognizing the inhumane experiments being carried out in his lab, Ansem ordered the experiments to end and sealed the laboratory. The experiments wouldn't end here; Ansem's students continued behind the back of their master, with Xehanort taking Ansem's name to continue his dark work. Ansem the Wise was banished to the Realm of Darkness by Xehanort, orchestrating a coup that left him leading the wise man's former students.
No longer under the scrutinous eyes of their former master, the students and their new leader dug deeper than before. They'd soon uncover the heart of Radiant Garden as well as the Heartless. Xehanort made the students into Nobodies, began manufacturing Heartless and split himself in two. These two forms would become the Heartless Ansem Seeker of Darkness and the Nobody Xemnas of Organization XIII.
Under New Management
While Ansem Seeker of Darkness had more direct plans to open Kingdom Hearts with a prophecy involving seven princesses of pure heart, Xemnas would go on to rally his former students (now Nobodies) to his side. Together they'd form Organization XIII and begin interfering with Sora and Riku's plans as soon as the duo defeated Ansem Seeker of Darkness. Splitting the group in two, Xemnas and his closest cohorts would take refuge on The World That Never Was while the remaining Organization members headed to Castle Oblivion to confront Sora and Riku.
In Castle Oblivion, a schism would form between members as new members, Marluxia and Larxene, sought to rewrite Sora's memories in an attempt to wrestle control of the Organization into their hands. They were opposed by senior members of the group also present, who'd use Riku and his affinity for Darkness to fight back. The Organization members stationed at Castle Oblivion's ultimate goals were to continue their studies on the heart, focusing on memories tied to the heart, progressing research that'd lead to the creation of data replicas clones.
These data replicas would serve Xemnas' purpose. He needed at least one Keyblade wielder to continue fighting Heartless, which in turn released hearts from the defeated monsters that Xemnas would use to power a device capable of creating an artificial Kingdom Hearts. When the members stationed at Castle Oblivion failed, Xemnas took it upon himself to antagonize Sora and his allies, spurring the young Keyblade wielder on to play into his plan unknowingly. When Sora became aware of the part he'd played in Xemnas' attempt to create a human-made Kingdom Hearts, he and his friends led an attack on The World That Never Was. Xemnas and his Organization XIII were defeated, but it was soon discovered that after having his Heartless and Nobody destroyed, Xehanort would return in his original form.
The Real Deal
While undergoing their Mark of Mastery exam to become full-fledged Keyblade Masters, Sora and Riku had their first encounter with Xehanort. He revealed that his master plan all along was to form what he'd call the "true Organization XIII." Intended to limit membership exclusively to 13 versions of Xehanort himself, the villain's plan included using the data replica research undergone at Castle Oblivion on top of time travel to pluck 13 versions of Xehanort from through time to wage the prophesized final battle between Light and Darkness. This fated encounter would re-forge the χ-blade, the key to opening Kingdom Hearts.
Sora and his allies would thwart Xehanort and his new Organization, but not before sacrifices were made. Though the Organization was defeated during the final battle between Light and Darkness, it hasn't stopped Xehanort from rearing his head again recently. That said, his appearance seems temporary, opening the series to whatever antagonist or group wants to take a shot at the Light and its defenders in the future.