dB
There's a mix of correct and slightly incorrect information above. Mostly correct, but ...
To be completely, totally, nitpickingly accurate, decibels are used to measure relative levels of power (if you don't believe me, look it up in a dictionary). So, say we measure the power of something (like a noise, say), and we want to say how "big" it is relative to something else. We can use decibels to describe the relationship: 10 * log10 (figure we're talking about / figure we're comparing it to) = decibels.
Decibels is a logarithmic scale. If a sound is 10 times more powerful than another sound, it's level is 10 dB greater. If it's 100 times more powerful, it's level is 20 dB greater. If it's 1000 times more powerful, it's level is 30 dB greater. If it's a tenth as powerful, its 10 dB smaller. If it's twice as powerful, it's 3.01 dB greater. If it's four times as powerful, it's 6.02 dB greater. If it 's half as powerful, its 3.01 dB less.
What does log10 mean? By definition log10 (A) equals the number that, when 10 is raised to it, is equal to A. In symbols:
log10 (A) = B means that 10^B = A.
So to put the definition of decibel another (perhaps more confusing way), the number of decibels is ten times the number that, when you raise 10 to it, equals the figure we're talking about divided by the figure we're comparing it to. Or, to say the same thing yet another way, if you raise 10 to the power dB/10, you'll get the figure we're talking about divided by the figure we're comparing it to. For example:
10 / 1 = 10^1 ... 1 x 10 = 10 ... 10 dB = 10 times the power.
100/1 = 10^2 ... 2 x 10 = 20 ... 20 dB = 100 times the power
1000/1 = 10^3 ... 3 x 10 = 30 ... 30 db = 1000 times the power
1 / 1 = 10^0 ... 0 x 10 = 0 ... 0 dB = the same amount of power
.1 / 1 = 10^-1 ... -1 x 10 = -10 ... -10 dB = 1/10th the power
2 / 1 = 10^.301 ... .301 x 10 = 3.01 ... 3.01 dB = 2 times the power.
4 / 1 = 10^.602 ... .602 x 10 = 6.02 ... 6.02 dB = 4 times the power
.5 / 1 = 10^-.301 ... -.301 x 10 = -3.01 ... -3.01 dB = half the power
Somewhat tricky part:
Voltage isn't a measure of power! Voltage is a measurement of electromotive force. Electromotive force (the "push" that makes an electrical current flow) transmits power (from the thing that creates the voltage to the thing that it runs through), but it depends on how much resistance there is in the thing that it runs through. For DC, power = voltage squared divided by resistance. Note that power varies with the square of voltage ... a doubling of voltage results in a quadrupling of power, for example.
Decibels are really used to compare power ... when we use the dB unit to compare two voltage levels, we're really talking about the power that those voltage levels will transmit in a given circuit.
So, by definition:
dB = 10 * log(Pm / P0) (P1=power we're measuring; P0=reference level, i.e. power that = 0 dB)
dB = 10 * log10 (Vm^2/R / V0^2/R)
Simplify the expression in parenthese slightly:
dB = 10 * log10 ( (Vm/V0)^2 )
Going back to the definition of "log10" for a second:
log10(A) = B means that 10^B = A. Thus, if
log10(A^2) = B, then
10^B = A^2, and
10^(B/2) = A.
So, if:
log10(A^2) = B, it follows that
10^(B/2) = A, and if that's true, reverse the definition and you see that
log10(A) = B / 2, which means that
2 * log10(A) = B
Put it all together:
log10(A^2) = 2 * log10(A)
Apply it to voltage:
10 * log10 ( (Vm/V0)^2 ) = 20 * log10 (Vm / V0)
for voltage:
dB = 20 * log10 (Vm / V0)
So, when we talk about voltage, the number of decibels is double what it is when we talk about power.
20 dB = 10 times the voltage
40 dB = 100 times the voltage
60 dB = 1000 times the voltage
-20 dB = 1/10th the voltage
6.02 dB = 2 times the voltage
12.04 dB = 4 times the voltage
-6.02 dB = 1/2 the voltage
Of course, none of this absolute -- it's just relative to some other figure (power, voltage) that we're comparing it to. To create an absolute measurement scale, we need to define a fixed level that we'll compare it to.
The two common references used to create an absolute scale for measuring voltage are:
dBV = .775V (1 mW power in 600 ohms)
dBu = 1V
+4 dBV = 1.2283 volts
-10 dBu = .3162 volts
People also have invented an absolute decibel scale to talk about sound pressure levels (SPL). SPL isn't power either -- like voltage it's the "push" (pressure) that transmits power. So the dB of a given SPL is also 20 * log10 (SPLm / SPL0). To create an absolute scale, people use a reference SPL that's equal to the threshold of hearing (i.e. the smallest sound you can year). So, when you go to a rock concert and the SPL is 100 dB, the SPL is 100,000 times the smallest SPL you can hear. What's perhaps more exciting is that it's 10,000,000,000 times more powerful.