What characterizes a second-order system and its response?

A second-order system has transfer function: G(s) = omega_n^2/(s^2 + 2*zeta*omega_n*s + omega_n^2), where omega_n is natural frequency and zeta is damping ratio. Response depends on zeta: underdamped (zeta<1) shows overshoot and oscillation, critically damped (zeta=1) is fastest without overshoot, overdamped (zeta>1) is slow without overshoot. Typical design targets zeta = 0.5-0.8 for good balance of speed and overshoot. Many physical systems are second-order or approximated as such.