When taking corners quickly, the biggest worry most drivers should have is slipping and losing control of the car. This happens when a driver takes the corner too fast. The physics of taking a flat corner depends on the equation vmax = Sqrt(mu*r*g). mu, the coefficient of static friction, is constant, as is g, the acceleration due to gravity. Therefore, a driver trying to take a corner as quickly as possible would like to make the radius of the turn as large as possible to allow for a higher vmax, keeping his car from slipping at higher speeds.
But how? Doesn't a road have a defined radius?
Yes, and no. The picture explains it. The arrow in the figure is what's called a "line" this is the best possible way for a car to take a corner at the highest speed. The line a regular driver would take is very curved, mimicking the road, and not allowing for a high vmax due to the small radius. A race car driver would take a better line. The racer's line is significantly less curved than the regular driver's line, making the radius much larger, allowing for a higher vmax . The racecar driver starts and ends wide of the inside and hits the apex of the turn, allowing for the least curved line possible.
To conclude, when trying to take a corner quickly, the driver of the car should start out wide, hit the apex, and end wide, causing a relatively high radius and a relatively high vmax, without having the car slip off the road.