My own non-scientific classifications of the various ways to produce sound:

Traditionally:
1) Hitting – causing an object to vibrate by hitting it with another object, whereby one of them is usually meant to vibrate for a longer period of time;
2) Plucking – to cause a disturbance to an object in rest (usually a string);
3) Bowing; Rubbing – using friction to cause an object to start to vibrate – using a bow or other slightly rough object;
4) Swinging through the air – so that the object swung generates a vibration in (and of) air;
5) Blowing – to create a column of vibrating air.

Special categories involving electricity:
6) Air set in motion through the use of loudspeakers or earphones;
7) Objects brought to resonance through the sound coming from loudspeakers – this can also involve feedback, whereby the resonance of the object gets picked-up, amplified and brought back to sound through the loudspeaker, causing the object to continue to resonate. A (or the) musical master of utilizing this feedback scenario in pop music is of course Jimi Hendrix a good example in the field of avant-garde music would be Alvin Lucier’s Music on a Long Thin Wire.

One more phenomena beyond this system of categorization would be vibrations applied directly to the organs of hearing that are able to give the same impression as that of hearing sounds. One way to test this for oneself is to cover both ears with the insides of the hands while placing the elbows on a surface that can be brought to resonance by someone or something else. The workings of the ear, the organ that picks up vibrations and translates these into neurological impulses that the brain can process further is part of the science of living organisms also known as biology.

Although nothing to do with the many ways sound can be generated but  something which will complete the story from sound generation to interpretation is psycho-acoustics or the study of the perception of sound. A fascinating branch of science that deals with the psychological and physiological impact of sound.