Dither is random noise added to a signal in order to increase the degree to which the quantization and sampling process accurately reproduces the image of the input signal. The simplest dither is quiet white noise, but more complicated forms of dither are possible using noise shaping, and they can even be completely inaudible. Signal details smaller than the quantization level are retained, after quantization, when Dither Noise is added to the source material.
A small signal detail with height less than the quantization level, and centred half way between two quantization steps, is too small to cross a quantization step. It is lost when the waveform is quantized. With a Dither Noise waveform added, the height of the two waveforms summed is occasionally greater than the quantization level. At these occasional points the waveform crosses a quantization step. So now when this waveform is quantized, although the result is noisy, the shape of the original signal detail is retained.
Bit depth reduction
A common use for dither is to retain the waveform shape defined in the bits to be lost when converting a digital signal from a higher bit depth to a lower one, e.g. from 24-bit to 16-bit.
When you reduce bit depth without dither there will be resulting distortion as well as the loss of the fine detail. This has two aspects, the change (error) in the waveform itself and an unpleasant addition to the sound because of the error. The waveform error is expressed in the output signal as noise, but noise related to (correlated to) the audio from which it is derived. This is a quality of noise that is generally found to be unpleasant.
When a small variation in the signal switches between quantization steps, it will be highly distorted, e.g. converted into a square wave. Dither will then reduce the distortion. Frequently flipping the least signifficant bit by the sum of the small signal and the dither noise retaines the shape that the waveform originally had. With the dither, the total noise is much less objectionable than the effect of quantization alone.
Going from 24-bit to 16-bit, the quantization error is small. Many sources (e.g. cassettes and LPs) already have considerable noise such as tape hiss. Even the best live recordings get some noise from the equipment, especially microphone preamplifiers. This might not make the best dither, but it acts in the same way.
HydrogenAudio is forum where examples of dithered conversions to 16-bit can be submitted for evaluating and blind testing.