Dithering

Analogue is infinitely continuous, digital is bits (zoom in on a file and see the steps; horizontal distance between samples is the sample rate, vertical distance between samples is the bit depth or "word length"). Higher sample rates mean higher frequencies can be represented. Higher bit rates mean better dynamic resolution.
Truncating the 8 LSB's from a 24-bit recording leads to 'quantization error', when the original waveform is compromised by a lack of resolution. This leads to noise, added harmonics and distortion.
Dithering adds low-level noise to the audio before the bit-rate conversion.
16-bit recording gives a maximum signal-to-noise ration of 96 dB. The noise floor is therefore -96 dB. A recording which has been dithered from 24 bits retains audible signals past the noise floor, therefore giving us a greater perceived dynamic range.
Dither noise is very low in level. It can be shaped so that the noise is shifted into specific frequencies.

HPF dither: Dither noise is in the high frequencies
"Near Nyquist": Dither noise is shifted up to near the Nyquist frequency (in a 44.1 kHz recording this would be around 22 kHz). This pushes the noise to the limits of our hearing.
"Psychoacoustic" noise shaping: Using the Fletcher Munson curves and the knowledge that dither noise is very quiet, we can see that the ear is most sensitive (at low volumes) at around 2.5-5 kHz and around 12 kHz. It therefore makes sense to attenuate the level of dither noise at those frequencies, which makes them much less noticeable to us.

Don't apply dither noise to a wave with DC offset
Don't normalise to 0 dB before dithering; leave at least 1 dB safety margin in case the dithering noise pushes the signal into clipping
Don't do anything (apart from final wave edits) after dithering
Only dither an audio file once. Go back to the original 24-bit file if you need another bit depth

The POW-r dithering algorithm, as used in Logic and many other software systems
Apogee's UV22 dithering algorithm
A definition of dithering and why we need it
The fabulous guide to dithering by Izotope (PDF)
A comparison of different dithering systems