Decibels are used to express acoustic and electrical level ratios.
A Bel (named after Alexander Graham Bell) is the logarithm (base 10) of the ratio of two measurements of power. Logarithms are useful in audio because our perception of loudness is logarithmic.
A decibel is one tenth of a Bel. 1B = 10dB and so the decibel value is ten times the logarithm of the ratio of two measurements of power.
For example, if a sound or electrical signal has twice as much power, it is approximately 3dB more powerful (because 10 x log 2 ~ 3). If it has ten times the power, it is 10dB more powerful.
Because a dB is a measure of power ratio, it does not tell you the absolute power. In order to use dB to express absolute measurements, you must also indicate what 0dB corresponds to.
For example, '3dB' on its own doesn't give you an absolute measure of power level but '3dB with 0dB = 1mW' does. In this case 0dB is defined as 1 milliwatt and so 3dB is a power level of approximately 2 milliwatts.
As an abbreviation of 'dB with 0dB = 1mW', we write 'dBm'. And so '3dBm' means a power level of (approximately) 2 milliwatts.
However, it turns out that in record engineering, we generally care more about voltage than power.
Because power is proportional to the square of the voltage, a voltage increase of ten times corresponds to a power increase of 100 times. So to use dB with voltage ratios, we must multiply by two. The decibel value is twenty times the logarithm of the ratio of two voltages. If an electrical signal has twice the voltage (assuming the same impedance) the (power) ratio will be 6dB.
To talk about absolute voltages, we need to define what voltage we mean by '0dB'. There are actually two reference voltages in use. 1V and 0.775V. When using the former as the reference, we write 'dBV' and when using the latter we use 'dBu'.
So '6dBV' means 2V and '6dBu' means 1.55V.
Voltage in this context is always root-mean-square (rms) voltage.
When we are talking about the level of a digital signal, 0dB is usually defined to mean the maximum level possible, i.e. the value when all the bits are set to 1. Because 0dB is defined at the maximum, digital signal levels will be expressed as negative dBs.
In the context of faders, 0dB just means a straight-through signal. Positive dBs indicate gain; negative dBs indicate attenuation.