Damping ratio controls oscillation behavior
A second-order system (like a mass on a spring with friction) responds to a sudden input (a step) in ways that depend on the damping ratio. Underdamped systems (low damping) overshoot the setpoint and oscillate around it before settling, like a car suspension bouncing. Critically damped systems (just the right amount of damping) reach the setpoint fastest without overshoot, like a well-tuned shock absorber. Overdamped systems (high damping) approach the setpoint without oscillation but take longer, like a heavily loaded suspension that settles slowly.
Choosing damping for the application
Control designers tune the damping ratio based on the cost of overshoot versus the cost of slow response. A pressure vessel regulator might be overdamped to prevent dangerous pressure spikes. A room temperature controller could be underdamped because slight overshoot and oscillation are harmless and response is faster. A manufacturing process might be critically damped to balance settling time and accuracy. The settling time (how long until the response stays within 2 percent of the final value) and peak overshoot are both functions of the damping ratio and natural frequency, so engineers adjust both to meet specifications.