The tendency for a tone to keep ringing after Attack. Many factors influence the sustain (rate of Decay) of a stringed instrument. Heavier-gauged strings tend to sustain longer, as do instruments made from denser woods (ash, maple, etc). Graphite instruments tend to have the longest sustain, as do instruments built with a longer scale-length.
Compressors are electronic effects which influence the sustain of a note by maintaining a consistent volume throughout the rate of Decay. Compressors, which "squish" the peaks of a sound wave, are often combined with boosts (which bring up the quieter parts of the wave) and noise-gates (which Cut off a note once it passes a pre-determined minimum threshold).
Sustain is not to be confused with "feedback," creating a feed-back loop in which the sound waves from a nearby speaker cause strings to vibrate, which in turn causes the speaker to deliver stronger sound waves, which in turn causes the strings to vibrate more forcefully, etc.