I'm not totally sure what your question is, but the -3 dB point is a convenient reference point for a change in power, rolloff of amplifier gain, rolloff of a filter at the band edges, etc.
Are you asking how it is calculated? The decibel is basically a power ratio described in logarithmic terms which makes it much easier to describe gain and changes in signal level than using linear descriptors in many situations.
The Bel is the log of a ratio of two power values (named after Alexander Graham Bell). Thus, a two-to-one ratio of power is LOG(2), or 0.3 Bels (actually 0.301029995 Bels, but it is seldom worth the trouble in conventional usage to carry the number of significant figures beyond 1 or perhaps 2 decimal places). To be more convenient, the decibel (dB), or 1/10th of a Bel, is more commonly used. Thus, a two-to-one power ratio is 10LOG(2), or 3 dB. Two uncorrelated signals of equal power added together (thus doubling the power) results in a 3 dB increase in signal level. (Things get more complicated when CORRELATED signals are added together and you get either cancellation or reinforcement, depending upon phase relationship, but we won't address that here.)
Since power is proportional to voltage squared, and multiplying by 2 is the logarithmic equivalent to squaring, we multiply the logarithm of signal ratios measured in volts by 2 in calculating decibels. Thus, a VOLTAGE ratio of 2 is expressed as 20LOG(2), or 6 dB. Same for current ratios or sound pressure ratios.) The "half-power-point," is 10LOG(1/2), or -3 dB.
Since the Bel or decibel is always a logarithm of a RATIO of two values, not an absolute value, a signal level expressed in decibels must either have the reference level stated along with the decibel value, or implied by common usage. This gives rise to the common units like dBv (decibels referred to 1 volt), dBm (decibels referred to 1 milliwatt), SPL (Sound Pressure Level referred to 20 micropascals), and many other commonly accepted usages of the decibel.
The reference values used in various areas of specialty is usually a convenient value for the specific specialty. Thus, 1 volt is a convenient reference value for people measuring signal levels with a voltmeter. 20 micropascals is the typical threshold of hearing perception for a young person, which makes it convenient reference pressure for sound measurements.
One other thing that might help; the 3db point is sometimes referred to as the "just noticable difference". A change in an audio signal of 1-2db can be difficult to hear, depending on what type of audio signal you're listening to, but 3db is the point (subjective) where most people can notice a change in the level. Incidentally, 6db (doubling of pressure - which is what our ears respond to) is the point where people say the loudness has doubled.
I assume, though I've never seen it explicitly stated, that with audio filters the 3db point is important because after that point the effect of the filter's attenuation can begin to be heard.
I have a minor quibble with disantlor's statement, "Incidentally, 6db (doubling of pressure - which is what our ears respond to) is the point where people say the loudness has doubled."
It is generally accepted that people subjectively interpret a 10 decibel change in sound pressure level as a doubling or halving of loudness. A 5 dB change is regarded as "very noticeable," but not a doubling.
I've heard that these rules of thumb on subjective perceptions of change in sound level were originally obtained from a large number of people at public events like county fairs being invited to listen to sounds in a controlled environment and tell the researchers how large they thought the change was. Of course, they had only their ears to judge by, and while ears are wonderful spectrum analyzers, they are not very accurate sound level meters. After all, mother nature designed them to operate over an enormous dynamic range from about 0 dB (for young ears) to about 120 dB (the threshold of feeling or tickling in the ear), or 140 dB (the threshold of pain).
20 hz to 20 khz is a different thing, it's a measure of frequency (subjectively heard as pitch). 20hz is theoretically the lowest pitched note we can hear and 20khz the highest. 0db is theoretically the quietest sound we can hear and 120dB the point where sound becomes painful.
did you have a specific question about audio frequencies?