If there is a video element in the show, whether it is moving projections or a character on a video screen, you need a Video Designer to design that. Sometimes if the designer is not also an editor, you need to pay a separate editor and maybe someone to program the cues in your theater so that it plays at the right time.
If the video involves a character speaking dialogue or a skit, you have Video Talent (if they are only on video and not live in the show, they are paid differently than the live actors) and a Video Director.
If the video involves footage owned by somebody else, you will need to pay them for it (Video Purchase) and maybe hire someone to edit down the footage to exactly what you need (Video Prep).
A Sound Designer is just what it sounds like: they design the sound for the show. That means they create whatever sound effects you need (telephones, doorbells, crashes, sirens, etc.) AND they are in the theater during tech adjusting the levels to make sure everything is heard and not overpowered or muddled together. When you have an orchestra, singers, sound effects, and tap dancing, not to mention an audience full of people crinkling candy wrappers, slurping drinks, and sucking up soundwaves, it is actually very difficult and takes a skilled designer to make sure the show sounds good in the theater.
A Technical Supervisor supervises ALL the tech on the show, from the stage crew to the sound and lighting designers, everything. He is there because most producers do not actually know or have the time to deal with the technical needs and crises involved in putting a show up, so the Tech Supervisor handles it.
Every show needs a Technical Supervisor and a Sound Designer. The video stuff is only needed if you have a video element in your show.
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