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The -3db bandwidth or the "3db down" bandwidth is the point or region between points where the response at those points is lowed by a ratio of 1/sqrt(2) which is approximately 0.7071 or more approximately 0.71 which is 71 percent of the center frequency response amplitude. This means that if you have an amplifier response that is -3db down (71 percent) at frequencies 100Hz and 300Hz, the bandwidth is 300-100=200Hz.
The choice for 3db comes from telegraphy if i remember right, due to the response of the human ear. However, regardless of how it started it is now in common use to characterize filter responses. For example, if you have two different filters one may have a smaller bandwidth than the other, and if one of your design goals is to achieve a sharp response, you would choose the one with the more narrow bandwidth. Conversely, if your goal is a broader bandwidth then you would choose the filter with the wider bandwidth. Even though these are not definitive absolute measurements for a given system, they help provide a method for comparison of filters and amplifiers.