Why logarithmic? The smallest perceivable sound level corresponds to an acoustic power density of approximately . But the level at which the sensation of sound begins to give way to the sensation of pain is about . To cope with this large dynamic range without loosing track of the number of zeros after the decimal point, a logarithmic scale is useful.
It's important to remember that a decibel measurement expresses a ratio. So it always makes sense to say that a signal x is so many dB greater (or less) than signal y. But if we say that a signal is equal to some number of decibels, then there must be a reference level. For sound, that reference level is usually taken as which corresponds to a pressure of . If we call this the reference pressure level, , we get the definition of sound pressure level
In a circuit, the choice of a reference level is not quite so obvious. For voltages, the typical choice is 1 V, which gives "decibels relative to 1 Volt" or dBV for short. Other forms you may encounter are dBW (relative to 1 watt) or dBm (relative to 1 mW).
It is sometimes stated that the response of a filter falls off at "20dB per decade" or "6dB per octave". This is just another way of saying that the response varies as 1/f. In other words, if is 10 times (i.e. the frequencies are separated by one decade) then will be 1/10 of . Since then we have a loss of 20dB (or a gain of -20dB) for each decade increase in frequency. Similarly, for two frequencies separated by one octave (a factor of two) we would have or approximately 6dB per octave.
For more information, check out the article on Decibels at UCSC Electronic Music Studios.