In simplest terms reverberation is the persistence of a sound in a space even after the source sound has stopped sounding. This persistence of sound is the result of the sound waves reflecting off of physical objects and walls in the environment. 
The reflected sound bounces back and forth among the reflective surfaces many times and creates a series of closely spaced echoes which increase in density over time to merge into a continuous sound.  These echoes decrease in intensity over time due to the absorption of sound energy by the reflecting surfaces. 
If there are no obstructions between the listener's ear and the sound source the direct sound will be the first sound to reach the ear. This is followed by a series of discretely separated echoes known as early reflections which provide cues as to the size of the space the sound occurs in with longer delays indicating a larger room.  The reverberation time, or RT60, is the time it takes the sound to decay 60 dB below its original level (or a millionth of that original level). Rooms with more reflective surfaces will result in longer reverb times and are refered to as "live" or "wet", while rooms with very absorptive surfaces will result in relatively short reverberation times and are refered to as "dead" or "dry".
  Sound Examples:  
voice (dry)  voice (reverb) 
The above example was created with a reverb simulator in Csound.

See Also:

reverb simulation    reverberators    digital delay    tape delay