A waveform is an exactly repeating pattern of vibration. Physically, sounds are described as waves which propogate outward from a source. These waveforms may be viewed on an ocilliscope, where one may see their shape.
For musical purposes, the most basic division between different types of sounds is whether or not they give the sensation of a particular pitch.
A sound which does not give a certain pitch -- i.e. most drums, a cymbal, a door slamming, etc. -- has a waveform whose shape is very complex and does not exhibit exact repetition, thus it is aperiodic.
A sound which does give a certain pitch, on the other hand, has a waveform which, no matter how complex it is, always repeats exactly a certain number of times per second, thus it is periodic. Almost all of the work done in tuning-theory concerns periodic sounds.
The length in time between one point in the periodic waveform and its later exact repetition is called its "wavelength". A sound's wavelength is inversely proportional to its frequency. Thus, a longer wavelength has a lower frequency, which means that it produces a lower pitch; and a shorter wavelength has a higher frequency, which produces a higher pitch.
Another measurement which may be made on a waveform is its "amplitude", which displays how loud the sound is. A graph of a waveform is generally plotted with time on the horizontal axis and amplitude on the vertical.
The simplest waveform, and the one which gives the "purest" sound, is a sine-wave. By "pure" we mean that there is only a single audible pitch, with no higher partials.
Most sounds used in music are complex tones, which means that a sound which is perceived as a single pitch is actually a conglomeration of a whole series of "pure tones", a fundamental with its partials.
These complex sounds produce differently-shaped waveforms. As more and more partials are added to a "pure" sound, the waveform transforms from a sine shape to a triangular "ramp" shape. Sounds which emphasize only odd-numbered partials (as a clarinet, for example) have a more square waveform shape.
Most acoustic instruments have waveforms with very complex shapes, but if they are periodic, the waveform will repeat no matter how complex.