Q:

Which unit does the power spectrum of microphone output have?

If I sample sound with a microphone and find the absolute square of the (non-normalized) DFT, I get the discrete power spectrum (correct me if I'm wrong). Which unit does it have?

A:

This depends a little on how thorough you want to be. Your microphone measures pressure and not actual power. That means your time domain signal has the units of Pascal and the spectrum has Pascal squared per Hz. In order to turn that into Intensity you would have to normalize by the free field impedance of air which is density*speed_of_sound and roughly 410 Newton seconds per cubic meter (at sea level & room temperature). Then the units would be Watt per square meter per Hz. You would only get to Watts per Hz if you integrate over a full sphere . This requires either an omnidirectional source or a rather tedious measurement involving a large number of microphone positions.
There is added complexity if you want to calibrate in dBSPL (sound pressure level). The 0 dB reference for this is 20 micro Pascal (which is close to the human threshold of hearing) and a pressure of 1 Pascal corresponds to roughly 94 dBSPL. 

A:

Power is measured in Watts ($W$). For a power spectrum the indiviudal bins have units of $W / Hz$.
However unless you calibrate you microphone/ADC input and eliminate other scaling factors (e.g. window compensation, factor of N / 2 in your FFT, implicit scaling in integer sample values, etc) then there is an arbitrary overall scale factor which needs to be accounted for if you really do want to measure absolute power levels.

