A waveform is an image that represents an audio signal or recording. It shows the changes in amplitude over a certain amount of time. The amplitude of the signal is measured on the y-axis (vertically), while time is measured on the x-axis (horizontally).
Most audio recording programs show waveforms to give the user a visual idea of what has been recorded. If the waveform is very low and not pronounced, the recording was probably very soft. It the waveform almost fills the entire image, the recording may have been too "hot" or recorded with the levels set too high. Changes in a waveform are also good indicators as too when certain parts of a recording take place. For example, the waveform may be small when there is just a vocalist singing, but may become much larger when the drums and guitar come in. This visual representation enables audio producers to locate certain parts of a song without even listening to the recording.