The term "staggered," when used in reference to wheels and tires, refers to having wider wheels on the back of a vehicle than are used on the front. Installing wider wheels on the back of a car is common on muscle cars, sports cars and super cars and is primarily used by manufacturers to provide additional traction at the rear while keeping the steering at the front light and responsive. Ferraris, for example, often combine wide rear tires with much narrower front tires. Installing staggered wheels is also a popular modification for cars that had the same size wheels and tires at all four corners as delivered from the manufacturer.
Cars that can take advantage of a staggered wheel package are generally powerful, rear-drive cars. The wider the tire can be on the back the better, since the power can be used without causing the wheels to spin. This is why horsepower-rich super cars like the Porsche 997 Turbo and Ford GT use a massive rear tire. This also helps handling, since it makes it easier to apply the power when exiting turns without causing the vehicle to power oversteer, which is where the back end comes around.
Lighter, More Responsive Steering
By fitting a narrower tire at the front, the steering of the vehicle can potentially be lighter and more responsive as well as provide more feedback to the driver. Wider tires make a car's steering heavier and can also dull the responsive and fine feedback that is delivered to the driver. By combining narrow front tires and wider rear tires, the driver can potentially have the best of both worlds: grip from the rear for putting down the power and feedback from the front so that she is aware of what the car is doing.
Another reason for installing a staggered wheel and tire package is the more aggressive appearance that wider rear wheels and tires give a vehicle. Because the rear wheels are wide, they usually come out to the edge of the wheel arch and in turn produce a better stance, which is how the car sits visually on the road.
Can Potentially Cause Unbalanced Handling
A potential downside to installing staggered tires is that the handling balance can become spoiled, particularly in a vehicle that does not have a chassis optimised for a staggered wheel and tire set-up. Wider rear tires provide more grip than narrower front tires, which can potentially cause understeer, which is where the front tires slide towards the outside of the turn and are in effect being "pushed" by the rear tires, which are sticking better to the pavement.
Cannot Rotate the Tires
A staggered wheel and tire package cannot be rotated, since the rears are side than the front. The wheels can only be switched from side to side on the front and back rather than from the back to the front. This can potentially have an impact on tire wear.