Tokyo Drift is a PS2 game released by Namco Bandai Entertainmment. It was released in 2006 and ties in somewhat with the film of the same name.
Tokyo Drift does not have a story as such. Players are challenged to beat all the racing teams in the city. The game starts off with you racing in Twinkie's Toyota Supra, which is borrowed. You beat a member of another team in a drag race which gets the ball rolling for you to continue the rest of the story. The game contains many drift races, however, the majority of gameplay is spent on the Wangan Highway, where you will generally take part in standard races, such as Point to Point events.
Tokyo Drift contains many Japanese cars, such as the Mazda RX-7 and the Honda NSX. These cars can all be customised, and upgraded to compete against different racing teams. Oddly, when the cars shift up, they will jerk quite violently regardless of how heavily upgraded they are.
While the game does not have a story, it does somewhat provide closure on the events of the film. One of the teams features Sean and Dom Toretto, who both appear in the film. The game can therefore be assumed to take place after the film.