The above answer ("Q is.....the voltage gain at resonance") is definitely wrong.
There is only one single definition: The quality factor Q is the so called "pole-Q" - defined by the pole position in the complex frequency domain (s-plane). The relation between the quality factor Q and the magnitude peak in the frequency domain for a 2nd-order lowpass/highpass is as follows:
Amax=(Ao * Q)/sqrt[1-(1/4Q²)] with Ao=DC gain.
For a bandpass filter the Q value defines the 3-dB-bandwidth of the circuit.
In the time domain, the Q value determines the step response as follows:
(1) For Q>0.5 the step response shows an overshoot "gamma" above the final value (when the transient has settled). This "gamma" value is given in % about the final value.
"gamma"=100 * exp[-3.14/sqrt(4Q²-1)]
Examples (gamma values in brackets): Q=0.5(0%); Q=0.7071(4.3%); Q=1(16.3%); Q=10 (85.4%
(2) The oscillatory decay is determined by the real part ("sigma") of the pole position only: exp(-|sigma|t).
The relation between "sigma" and the Q value is |sigma|=wp/2Q with wp=pole frequency.