As posted, downforce is a downwards force on a car using aerodynamics. The purpose is to increase the downforce in the tires, which increases how hard a car can take a turn. A FIA F1 race cars generates about 1g of downforce at around 115mph, and about 2g's of downforce at 163mph. At 163 mph, this translates into about 4g's of cornering force, and because there's about 1g of aerodynamic drag, 5g's of braking force.
There are serveral ways cars generate downforce.
FIA F1 cars use the upper body and wings. The rules state that the cars must run skidboards, so no underbody tunneling is allow. Champ cars (USA) and Indy Racing League cars are allowed to use underbody tunneling.
NASCAR cars tilt the body downwards (rear is about 1.5 inches above front), and use an air dam to block air flow from entering from in front of the car. (This should drive Bernoulli fans nuts, the slower moving air (relative to the car) underneath the car has less pressure than the faster moving air above. It's because the air dam is doing work on the air behind the dam, accelerating it forwards.) The air does flow in from the sides, so a rear spoiler is used to add downforce to the rear of the cars.
Other methods have been used as well. Older F1 type cars had side skirts to prevent air from flowing underneath the car from the sides (the wing up front directed a lot of the air upwards). The high amount of downforce was rough on the drivers and prone to accidents if a skirt was damaged, so they banned them.
Chaparral used large fans at the back of a solid body Lemans type car to draw air out from under the car, but this was later banned.
Might as well post a video of a F1 car in action. David Coulthard in a 2002 F1 McLaren at Spa.
spaf1.wmv