In a damped oscillation, the amplitude decreases exponentially as $A(t) = A_0 e^{-\gamma t}$. The time after which the amplitude becomes $\dfrac{A_0}{e}$ is:

Setting $A(t) = A_0/e$ in $A(t) = A_0 e^{-\gamma t}$: $\dfrac{A_0}{e} = A_0 e^{-\gamma t} \Rightarrow e^{-1} = e^{-\gamma t} \Rightarrow \gamma t = 1 \Rightarrow t = \dfrac{1}{\gamma}$. $1/\gamma$ is called the **characteristic decay time** of the damping. Energy decays at twice that rate ($E \propto A^2$), so energy falls to $1/e$ in time $1/(2\gamma)$. The quality factor $Q = \omega_0/(2\gamma)$ ties this damping rate to the number of oscillations before energy drops to $1/e^{2\pi}$.