When Alexander Graham Bell (harbinger of telemarketers and your future brain tumor) invented the telephone, he needed a simple way to deal with the changes in power that occur from transformer to transformer. Hence the Bel, commonly expressed in decibels (dB), a logarithmic ratio of change between two signals. The decibel, as it turns out, can also be used as a handy dandy reference with numerous applications, especially in the fields of electronics and audio, and also plate tectonics.

The decibel, as the unit of acoustic pressure that it is most widely refered to, is a logarithmic ratio of pressure, also known as dB SPL (or Sound Pressure Level). An increase in 10 dB SPL is an increase of pressure by a factor of 3.16. Sound is pressure. However, the way we perceive sound, a difference of 10 dB SPL is only twice as loud, so a nearby jackhammer (140 dB SPL) is 64 times louder than a standard dial tone (80 dB SPL). The range of human hearing is commonly understood to be between 0 dB SPL (the threshold of hearing) and 120 dB SPL (the threshold of pain), though obviously we can hear sounds above 120 dB SPL (just not very comfortably or clearly). Negative sound pressure levels exist, but humans won't be able to hear them until 5242 AD.

Decibels are calculated thusly:

For power (in Watts), dB = 10 log10 (P2/P1).
For voltage and pressure, dB = 20 log10 (P2/P1).

So a doubling of power is 10 log10 (2/1) = +3.01dB
and a halving of power is 10 log10 (1/2) = -3.01dB
and a tenfold increase of power is 10 log10 (10/1) = +10.00dB
and a change of power by one tenth is 10 log10 (1/10) = -10.00dB
and you can also figure that 5 times power is roughly +7dB (because you're halving a tenfold increase, right?) and 1/5 is -7dB and so forth.

Decibel references are also important. Because saying something is operating at +4dB is effectively meaningless (as opposed to saying that a +4dB gain is present), it is useful to have a couple of constants to compare levels. A decibel reference is a way to understand the relationship of power or pressure levels.

Here is a list of common references, where the reference is substituted for the input in the decibel equation:
0 dBW = 1 watt
0 dBm = 1 milliwatt
0 dBV = 1 volt
0 dBu (or dBv) = 0.775 volts
0 dBµ = .000001 volts (1 microvolt)
0 dB SPL = .00002 pascals

Different references have different uses. A common one (dBu) is used to explain the voltage of electronics independent of the load. For example, professional audio equipment runs at a nominal operating level (0VU) of +4dBu, or 1.228v (20 log10 1.228v/0.775v = +4dBu) and consumer audio equipment runs at a level of -10dBu, or 0.245v (20 log10 0.245v/0.775v = -10dBu).

We could also tell that a microphone which has an overload limit of 130 dB SPL starts clipping when exposed to 63.24 pascals (and that 1 pascal is about 94dB SPL, which is also about where, according to OSHA, sustained exposure causes hearing loss).

When adding references (say 7dBV + 3dBV), each individual reference must first be extrapolated into its base unit (in this case, volts, so: 7dBV = 2.24v and 3dBV = 1.41v) and then added (3.65v) and then recalculated into the reference (20 log10(3.65v/1v)=11.25dBV). They may not simply be added together.

Decibels are handy if you want to gauge what is happening in an audio system due to changes in the fader (or volume knob or any variable amplifier for that matter), or need to know how much power you're going to be sending through your system after a gain stage (or anywhere through any electrical system).