What effect does bit depth of an ADC have on noise floor, THD, or any other specification?
A analog to digital converter (A/D) takes the continuous analog input signal and produces a number that quantizes it at best to the nearest of a bunch of pre-determined levels.

Let's say you can give me any real number from 0 to 100, and I quantize that spread to the nearest one of a few discrete levels we picked up front.  To make the illustration more intuitive, let's consider a 2 bit A/D.  That means it can only produce one of 4 possible output values.  To minimize the worst case error, I divide up the 0-100 input range into 4 equal bands, 0-25, 25-50, 50-75, and 75-100.  I report the input value as either 0, 1, 2, or 3 depending on which of these bands it is in.  This is precisely what a A/D does, at best.  The difference is that most have more than 2 bits, and therefore have more bands to break the input range into.  In general, the number of bands is 2n, where N is the number of bits.

Now look at what error that adds.  If I report a value is in band 0, all you know is that it is from 0 to 25.  The best you can do is assume 12.5, and you know you'll be within 12.5 of the original.  Put another way, 12.5 is the amount of uncertainty on each measurement.  By assuming the input was in the middle of each band, you essentially have to assume there is up to 12.5 of noise on the signal.  Related to the full input range, the noise is therefore 12.5/100 = 1/8.  This is assuming a perfect A/D, limited only by the number of unique numbers it can produce.  The inherent noise level of the digitally encoded signal is therefore 1/8, which is 18 dB.  This particular unavoidable noise due to quantization only is called, surprisingly enough, the quantization noise.  In general, it is 1/2n+1.  Expressed in dB, is is pretty close to 6(n+1) dB.

There are other noise figures for a A/D, like how close the overall input to output curve is to the ideal straight line (linearity), and what the worst case error on the size of any band is.  These get into quality of the specific A/D and are not inherent to the process like quantization noise is.

Added:

As pointed out in a comment, the 6(n+1) dB noise figure is not the only story.  If you are making individual uncorrellated measurements, this is the maximum possible noise, and generally what you have to work with since you don't know that it doesn't apply to any one measurement.

However, if you are sampling a continuous signal that is band-limited to less than half the sampling frequency, then you probably care more about average noise.  In that case, you consider that 1/8 of full scale is the worst case error on any one reading, but statistically errors on individual readings will be linearly spread between 0 and 1/8.  Taking the RMS of such a noise signal results in a significantly lower effective noise level than the worst case I computed above.  It looks like Rawbrawb has gone into some derivation of this noise, so I won't go further.

Which noise figure you need to use to design your system depends on system level issues that we don't know here.  The point is to beware there are different legitimate ways of looking at the quantization noise, and you have to think carefully what each individual measurement means to decide which view is applicable to the error or noise analisys of your system.