White noise is a pure white noise. It should be noted that it has no particular dominant frequency. Thus, sounds that use this waveform will sound the same at all frequencies. For this reason, and to clear out the display, it is recommended to place those in the low ranges of the frequencies (or at the bottom of the keyboard notes), to keep the display organized. The actual frequency of the sound has strictly no impact on how this sound will be heard.
PinkNoise is similar to White noise, but its output is filtered in the high frequencies to produce a pink noise.
The Random waveform is an experiment at producing randomness in the waveform that is tied to the current frequency of the linked sound. It has little true "audio" meaning. It can be used to generate LFO amplitude modulation, or produce a wide variety of noises. The higher the frequency of the sound, the more closer the "random" waveform will come to produce actual white noise.
The Random Triangle is somewhat similar to the Random waveform, has the same uses, but its basic shape is triangular rather than square, reducing possible audio artifacts such as "clicks" when it used as a low frequency modulator.
The Constant setting just output always one. It can be used to modulate amplitude or frequency according to the envelope rather than the waveform.
The Silence setting just output always zero. It can be used to extend patterns with a starting or ending muted silence.
The Periodic noise is an harmonic waveform whose harmonics are randomly set. It has 2 parameters: Triangle or Square parameter sets the shape of the periodic noisy waveform. While the intensity parameter sets the size of the random noise table used to generate periodic noise.
The "Crazy" waveform is another example of a non-harmonic waveform.