OK - I think I got it now (sorry about some of the misinformation above).
I think this site clears up a lot of it (with examples):
http://www.phys.unsw.edu.au/jw/dB.html
deciBel = 1/10th of a Bel (Bel = Created by Bell labs to quantify the reduction in audio level over a 1 mile length of standard telephone cable).
1 db change = Just Noticeable Difference between sound levels. It's not easy to hear the difference between 2 sounds that vary by 1db (check out the examples in the link above).
3 db change = Noticeable Difference between sound levels. I always heard that anything below this was not 'perceived' as a volume change by the average ear. Doing more reading though - I think this is subjective.
3db increase = roughly double power
6 dB increase = approximately four times the power and twice the voltage
10x power = double the volume
Half the power = reduce the volume by 3db