In a ground vehicle with a suspension, the unsprung weight (or, more properly, the unsprung mass) is the mass of the suspension, wheels or tracks (as applicable), and other components directly connected to them, rather than supported by the suspension. (The mass of the body and other components supported by the suspension is the sprung mass.) Unsprung weight includes the mass of components such as the wheel axles, wheel bearings, tires, and a portion of the weight of driveshafts, springs, shock absorbers, and suspension links. If the vehicle's brakes are mounted outboard (i.e., within the wheel), their weight is also part of the unsprung weight.
Pneumatic or elastic tires help by providing some springing for most of the (otherwise) unsprung mass, but the damping that can be included in the tires is limited by considerations of fuel economy and overheating. The shock absorbers, if any, damp the spring motion also and must be less stiff than would optimally damp the wheel bounce. So the wheels execute some vibrations after each bump before coming to rest. On dirt roads and perhaps on some softly paved roads, these motions form small bumps, known as washboarding or "corduroy" because they resemble smaller versions of the bumps in roads made of logs. These cause sustained wheel bounce in subsequent vehicles, enlarging the bumps.
High unsprung weight also exacerbates wheel control under hard acceleration or braking. If the vehicle does not have adequate wheel location in the vertical plane (such as a rear-wheel drive car with Hotchkiss drive, a live axle supported by simple leaf springs), vertical forces exerted by acceleration or hard braking combined with high unsprung mass can lead to severe wheel hop, compromising traction and steering control.
Though this is usually not considered important, at least in the popular literature, there is a positive effect. High frequency road irregularities, such as the gravel in an asphalt or concrete road surface, are isolated from the body more completely because the tires and springs act as separate filter stages, with the unsprung weight tending to uncouple them. This can improve overall safety.
Inboard brakes make a big difference, but put more load on half axles and (constant velocity) universal joints and require space that may not be easily accommodated.