Wheel bushings are among the simplest to understand of all of your car's components, primarily since it doesn't have them. Bushings perform an entirely different task than the bearings that they're often confused with, acting more to isolate shock and vibration than to enable rotational movement.
Bushings are isolators generally made of rubber or a synthetic substance such as polyurethane. The inside of a bushing contains a metal sleeve through which a bolt may pass. Manufacturers insert these sleeves into a thick cylinder of rubber, and that cylinder slides inside of a chassis mounting point.
You can find bushings at nearly all of your suspension's articulation points. These include the control arms where they attach to the chassis, the anti-roll (anti-sway) bar mounts and at various points on the steering linkage. Components such as the upper ball joints work similarly to a bushing, but aren't bushings themselves.
Like anything else made of rubber, bushings will wear out and soften over time. Worn bushings allow the suspension to flex and move in ways that it shouldn't, which is a major contributing factor to sloppy steering and bad handling in older cars. You can replace the bushings with new rubber units, or you can replace them with polyurethane units for increased performance. Hard poly bushings will greatly increase your car's steering precision and handling, but do so at the expense of extra vibration and a decrease in ride quality.