Short of its wheels and tires, a car's struts are the single largest and most important part of its suspension system. The struts are essentially combination shock absorber and spring packages (also called a coil-over strut) with a large leg on the bottom connected to the wheels' steering hub. Struts act as the upper control arm in the car's suspension system.
Excess Body Roll
Although the car's anti-roll (aka "sway" bar) does help to prevent excess roll, the primary body control agents are its main springs. During use, a spring converts a certain amount of the movement it absorbs into heat, which slowly tempers the metal. After many thousands of miles, the metal becomes softer and more flexible than it originally was, which causes excess body roll while cornering or during steady-state cornering. A car with four completely worn springs will tend to "float" over minor imperfections in the road surface. The end result is a car that handles something like a barge in the ocean.
Uncontrolled Wheel Movement
The other end of the strut spectrum is the shock absorber, which is responsible for slowing the wheels' up and down movement. Without shock absorbers, the car's spring-equipped wheels would act like a super-bounce rubber ball dropped onto pavement; each successive bounce and rebound cycle would be a little less energetic than the last until it finally settles onto the ground. This phenomena will first express itself as a slight instability while travelling down rutted roads, and will eventually result in a severe "wash-boarding" up-and-down bounce over minor imperfections or while accelerating.
Because shock absorbers are responsible for slowing the wheels' up and down movement (and the car's body's tendency to roll while cornering), blown shocks are most noticeable during quick left-right-left transitions. A car with blown shocks will tend to continue leaning one way while the car is actually travelling the other. The worst case scenario is that the body roll meets an harmonic convergence with steering input and either rolls over or spins out.
One major symptom of worn struts is bottoming out over large potholes and speed bumps. Bottoming out is often a symptom of blown shock absorbers but can also be indicative of soft springs. Either way, the spring and shock assembly is a unit, so consider the whole thing shot.
Many modern cars use electronic shock absorbers that self-adjust to road conditions. They are designed to become softer over large imperfections and harder during spirited driving over smooth surfaces. If your car suddenly begins to exhibit symptoms of overall body roll or nose-dive under braking, or suddenly develops a very hard ride, then the strut system's electronic controls may be malfunctioning.