Primarily found on a rear-wheel drive vehicle, staggered wheels are when a car's wheels and tires are different sizes on the front axle than they are on the rear axle.
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Having a wider setup in the rear allows for more tire contact with the road, increasing grip and acceleration. This is why you tend to see the high-power street cars running a thick rear wheel and tire setup. They need the extra width to be able to put all of that power to the ground.
There are a few reasons people run a staggered setup. First, it looks killer! Having a staggered setup creates a far more aggressive stance on a lot of vehicles. It's not uncommon to see this style run on cars like the Mustang, E55 AMG, and Supra. A lot of these cars have wide rear hips that require a wider wheel to get proper fitment.