An example would be when you're reducing bit depth from 24 bit to 16 bit with no dithering.
Here's the process for each sample FYI:
1) Express 24-bit value in 16-bit integer range (two's complement), which is 1 sign bit, 15 integer bits, and 8 fractional bits.
2) Round the 8 fractional bits to the nearest integer. This is done by simply adding 128 (which is 0.5 with the sample expressed as indicated in step 1) to each 24 bit value.
3) Take the upper 16-bits of the 24-bit value to use as your 16-bit sample.
So for periodic signals with frequency components at divisors of the sampling frequency (Fs/4, Fs/5, ...) this same rounding will occur each period of the signal. This adds non-random errors which will be non-white noise and has the property of having most it's energy at signal harmonics and intermodulation frequencies.
This generally isn't much of an issue with music which is not simple periodic signals, but it theoretically does add distortion along with the noise.
Best of all, dithering eliminates this problem. You just add a small amount of random noise energy to each 24-bit sample before doing the rounding procedure above. An RMS of half a bit (so 128 on average) completely eliminates the issue. Even 0.25 (64) works really well.