The canonical definition of dB is in terms of a _ratio_
of the _power_ levels.
A Bel is a factor of 10 in the power.
10 dB is a factor of 10 in the power.
This is the gold standard.
It measures relative (not absolute) power.
============================================
Descended from this are multitudinous derived units. For starters,
let us consider some absolute measurements:
dBm measures power relative to the 1 milliwatt level.
That is, 0 dBm is (by definition) 1 mW.
The dBm therefore measures absolute (not relative) power.
This is a precise and sensible unit in microwave work, but sometimes
abused in consumer and semi-pro audio work.
dBA is the A-weighted power. You run the signal through a particular
filter that represents the "standard" human auditory system, and measure
what comes out on some absolute scale. (I'm not quite sure what is the
physical basis of the 0dB reference calibration is in this case.)
In my opinion, it is ungrammatical to use dB (without any suffix) to
describe any absolute power scale ... but alas there are millions of
people who don't take my advice in this matter.
There are also various ways of using dB to measure voltages; for a linear
system under *some* conditions, 20dB is a factor of 10 in the voltage.
But remember dB is _fundamentally_ a power unit, so extending it to measure
voltage only makes sense if the impedance (and other things that affect the
power-versus-voltage relationship) are held constant.
Things go from bad to worse when folks attempt to use dB to measure voltage
on an absolute scale.
WHEN IN DOUBT, USE dB TO MEASURE POWER RATIOS. That is the original, canonical,
fundamental definition.