First of all, when using or abbreviating a unit named after a person (in this case, Alexander Graham Bell), it is customary to capitalize the abbreviation, so the most respectful way to write the common measurement of the ratio between to values is dB. Capitalization is not used when the unit is spelled out with a scale prefix ("deci-", in this case), but the unit written out without a scale prefix is capitalized: "Bel".
Almost always, decibels are measured and reported based on a reference value, and the reference value is indicated in the unit abbreviation by extra characters added on after "dB".
Examples (the first one being the most common answer to your question):
- dB SPL (sound pressure level) – for sound in air and other gases, relative to 20 micropascals (μPa) = 2×10^(−5) Pa. Very frequently, dB SPL measurements of wideband sounds (like music, as opposed to a single frequency) are measured as an average across frequencies and then weighted to approximate the response of human hearing. The most common weighting is called "A-weighted". Sometimes this will be abbreviated dB(A). Note that SPL measurements are pressure and not power levels.
- dBm – electrical power relative to 1 milliwatt.
- dBu – RMS voltage relative to approximately 0.7746 V.
- dBV – voltage relative to 1 volt, regardless of impedance.
- dBFS (Full Scale) – the amplitude of a signal compared with the maximum which a device can handle before clipping occurs. This is used for levels in digital audio when samples are composed of signed integers of fixed bit depth. If the level were referenced to integer 0, different bit depths would have different level measurements. By using dBFS, levels can be directly compared between different digital formats. An absolute value of 32,767 in a 16-bit file and an absolute value of 8,388,607 in a 24-bit file would both be full scale and therefore be represented by 0 dBFS. Changing the MSB from a 1 to a 0 in each of these values would halve the total integer value and represent a drop of 6 dB, which would be a level of -6 dBFS in both cases.
Important Note: All of the above values are actually "20-log" dB measurements except the dBm and the dB SPL, which means the formula for calculating the dB based on the reference value is actually 20log(m/r) (m being measured value and r being the reference value). Power levels are scaled by 10 times the logarithm of the ratio, while intensity or voltage levels are scaled by 20 times the logarithm of the ratio. I'm not exactly sure why this is done, but I do know that it helps when calculating gain and output, etc. That's because, for example, in an electrical system power and voltage are not linearly related. If you halve the voltage into a given load, you reduce the power output to a quarter of the previous value. By using the 10-log and 20-log systems, half the voltage is a 6 dB drop and one quarter the power is also a 6 dB drop. The mixed scales make it so following gain changes through a system is much easier than if a consistent scale were used for both power and intensity.
Attribution: Portions of this answer were quoted from the Wikipedia page for "decibel".
I just noticed (while editing your question to make it more clear) that you specifically asked about acoustic power levels. Acoustic power is measured and reported in straight-up Watts, and is also measured in dB referenced to 10^(-12) Watts. Acoustic power is not often used in musical contexts because it doesn't relate as well to how we actually hear. One way to understand this is to note that the reference value for dB SPL is the average/typical measured threshold of hearing for humans. That means a sound pressure of -3 dB SPL should be inaudible to almost all humans and an pressure of 3 dB SPL should be barely audible to almost all humans. Sound power levels are not as immediately helpful in estimating the audibility and loudness of a sound.
The other difference between sound power and sound pressure is that sound power is meant to be the total output of a sound producing object, while sound pressure is the actual acoustic pressure measured at a particular point in space. As our ears exist at a particular point in space (or two points) and our ears respond to pressure and not power, sound pressure makes more sense for understanding how a sound will be heard. Sound power is more useful in engineering situations where acoustic power can effect physical objects other than human eardrums.
Also see the Wikipedia page for "sound power".