Turbines are simpler than piston engines, containing fewer parts, and are commonly associated with jet aircraft. They have a long history of use in electrical power generation, but haven't really made their way into vehicles in any large measure. Their most famous appearance under the hood came in a limited production run of Chryslers in the early 1960s.
Chrysler began experimenting with turbine power in the mid-1950's and the Turbine Car was actually the company's fourth-generation vehicle. Fifty of the cars were deployed for consumer testing, with more than 200 drivers logging three months behind the wheel. The four-person coupes resembled Ford Thunderbirds, and were fitted with turbine engines mated to a conventional drivetrain including a TorqueFlight automatic transmission. Chrysler rated the Turbine Car at 130 horsepower and 425 lb-ft of torque, and it could run on unleaded gasoline, diesel, kerosene or jet fuel.
Other notable turbine-powered vehicles included a series of GM Firebird concepts – not related to the Pontiac to later take that name -- developed in the 1950s. General Motors also built a special prototype of its EV1 electric that used a trunk-mounted, 40-kilowatt turbine in 1999.
For more details on how turbine engines work, hit up Wikipedia: http://en.wikipedia.org/wiki/Turbine
For more on the Chrysler turbine car, check out http://turbinecar.com/