Buck Bokai was the persona adopted by a non-corporeal species who traveled from the Gamma Quadrant through the Bajoran wormhole and had taken a position in the plasma fields near the Denorios belt, where they read as elevated thoron emissions in 2369.
That year, this Bokai persona was taken from the imagination of Jake Sisko, whom he followed home from the holosuite, after he had helped Jake work on his fielding techniques. Commander Benjamin Sisko then brought Bokai to the operations center and introduced him to the senior staff. When they tried to explain his existence, he simply said, "Don't look at me. I can't figure it out either." After recalling the events that lead him to the Sisko's quarters, he explained he and Jake were "getting hungry". This surprised Kira Nerys, who had never heard of "a hologram with an appetite." Bokai, had no idea what she meant by "hologram", but after a series of tricorder scans, Julian Bashir assured Kira that "he's quite real."
As the alien continued to feign believing that he was the real Bokai, the crew began to compare notes, and realized that the three individuals they were interacting with, Bokai, Rumpelstiltskin, and Jadzia Dax, were all products of someone's imagination. Bokai continued to pursue Sisko and discuss the topic of baseball only to be a bit discouraged by Sisko's desire for him to "disappear for good."
The three aliens later converged in the replimat to discuss their findings, only to be confused by the irony of their results. Bokai, however, disagreed with the other two's desire to "abandon the whole thing," explaining that he believed he made a connection with Sisko, as he sensed a curious feeling regarding Sisko's affection for Bokai.
Following the crew's realization that their own imaginations were responsible for the events that transpired on the station, the crew was finally able to return things back to normal. After returning to his office, Bokai revealed himself a final time to justify their actions, firstly explaining that "this imagination of yours, it's a tough concept for us to get a handle on," before explaining that "we're on an extended mission exploring the galaxy. Awhile back, we followed one of your ships through that "hole in space" out there. We've been watching you, trying to figure out the rules of your game." When asked why they didn't just make direct contact like he now was, Bokai explained that, "we've learned you never can tell how someone will treat the visiting team," and that "that's what this has been all about. Learning to understand."
At the conclusion of their conversation, Bokai explained that they had to go, but when Sisko protested that "you haven't told us anything about your species," Buck simply replied, "I'd like to. Maybe next year." (DS9: "If Wishes Were Horses")