Tanner, an NYPD detective gone undercover due to his impressive driving skills, must gain the confidence of the syndicate's bosses by performing increasingly difficult missions such as stopping or following another car, driving through windows, delivering a stolen car or scaring a taxi customer. The game is played in four cities—(Miami, San Francisco, Los Angeles, and New York City)—which, like many games, remain only partially faithful to the original city layouts but notably featured them as explorable open world environments.
The game was designed to mimic '60s and '70s car chase movies. The title and the basic theme seem to be heavily inspired by the 1978 film The Driver by director Walter Hill. Most notably, the Training level at the beginning of the game is a copy of a scene in the movie in which the Driver proves his skills to some gangsters in a parking garage. The music, overall character design in the cut scenes, and the cars themselves (complete with flying hubcaps), are inspired from movies such as Bullitt and even TV shows such as Starsky and Hutch. The game also features a Director's Mode, where the run could be replayed with cameras chosen by the player, and a Quick Replay, where the cameras were automatically selected.
In addition to the Undercover mode, the player also had an option to drive around in the cities (once they were unlocked) and dirt tracks. There were several other modes, like pursuit, escape, checkpoint, and carnage (inflicting as much damage as possible).
Source: Wikipedia, "Driver (video game)," available under the CC-BY-SA License.