Fizzbin is a fictional playing-card game created by Captain James T. Kirk in the Star Trek episode #49 "A Piece of the Action".
The game was "invented" while Kirk, Spock, and Dr. McCoy were being held hostage on Sigma Iotia II. Kirk spontaneously created a confusing card game to distract the henchmen guarding them. When he claimed that the game is played by inhabitants of the planet Beta Antares IV, Spock replied that he is familiar with the inhabitants; Kirk quickly cut him off to prevent his science officer from professing ignorance of the game.
Beta Antares IV uses slightly different cards, but a standard Earth deck of cards will do. The rules are intentionally very complex. Each player gets six cards, except for the player on the dealer's right, who gets seven. The second card is turned up, except on Tuesdays. Kirk dealt the henchman two jacks, which are a "half-fizzbin." When the henchman said he needs another jack, Kirk warned that a third jack is a "shralk" and is grounds for disqualification. With two jacks, one wants a king and a deuce, except at night, when one wants a queen and a four.
At this point, Kirk dealt a third jack, but to keep the ruse going, he ignored the disqualification rule he had just made up. He explained that, had a king been dealt instead of a jack, the player would get another card, except when it's dark, in which case he'd have to give it back. The top hand is a "royal fizzbin," but the odds of getting one are "astronomical": when Kirk asked Spock what the odds are, Spock truthfully replied that he had never computed them.
Kirk called the last card a "kronk" and then purposely dealt a card such that it fell on the floor. As the henchman being taught reached down, Kirk delivered a blow to his neck, and Spock nerve-pinched the other guard, allowing the three to escape.
Quark mentioned the game in the Star Trek: Deep Space Nine episode "The Ascent" as a way for him and Odo to while away the time while trapped on a runabout. (Since it was unlikely he had heard the specific story of Kirk's ruse or was intending to swindle Odo, this implies that such a game actually exists in the Star Trek universe. Either Kirk used the name of a real game when making up his imaginary one, or the Iotians -- shown to be highly impressionable -- later invented a game based on Kirk's rules and spread it through the galaxy. A third possibility is that fizzbin is a fictional game used by tricksters throughout the galaxy to deceive others who are unaware of the game.)