This depends on the area of interest.
In resonant circuits, impedance bandwidth is mostly the difference between the frequencies where the impedance increases with factor 1.414 (series resonant circuit), or decreases with factor 1.414 (parallel resonant circuit). BWimp = Fcenter/Q. In antennas it can be the difference between the frequencies where the VSWR reaches a certain value (for example SWR=2).
Power bandwidth of a system is mostly the difference in frequencies where the power dropped with a certain percentage below the mid-band output power. Half-power bandwidth is frequently used.
In signal analysis, the power bandwidth can be that frequency bandwidth that contains a certain part of the total available power (for example 50% or 99%). This definition can also be applied to filters that have noisy input signal. A lossless filter that outputs 1uW based on an input signal of 1nW/Hz has a power / noise bandwidth of 1u/1n=1000 Hz.