Why logarithmic? The smallest perceivable sound level corresponds to an acoustic power density of approximately \(10^{-12}~\text{W}/\text{m}^2\). But the level at which the sensation of sound begins to give way to the sensation of pain is about \(1~\text{W}/\text{m}^2\). 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 \(x\)'s amplitude is so many dB greater (or less) than signal \(y\)'s amplitude. 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 \(10^{-12}~\text{W}/\text{m}^2\) which corresponds to a pressure of \(10^{-5}\text{N}/\text{m}^2\). If we call this the reference pressure level, \(p_0\), we get the definition of sound pressure level $${\textrm SPL} = 20 \log\left(\frac{p}{p_0}\right)$$ (where we use 20 instead of 10 since power is proportional to the square of the pressure).
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 \(f_2\) is 10 times \(f_1\) (i.e. the frequencies are separated by one decade) then \(\vert H(f_2)\vert\) will be 1/10 of \(\vert H(f_2)\vert\). Since \(20 \log(1/10) = -20\) then we have a loss of 20 dB (or a gain of -20 dB) for each decade increase in frequency. Similarly, for two frequencies separated by one octave (a factor of two) we would have \(20 \log(1/2) = -6.02\) or approximately 6 dB per octave.
For more information, check out the article on Decibels at UCSC Electronic Music Studios.