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For questions specific to aero-/tri-bars including clip-on extensions.

Wikipedia sums them up better than I can:

Triathlon bars or aerobars include various styles of aerodynamic handlebars for racing bicycles and particularly time trial bicycles. Included are narrow, bolt-on extensions that draw the body forward into a tucked position, pursuit bars that spread the arms of the rider but drops the torso into a slightly lower position, and integrated units that combine elements of both designs.

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Triathlon bars are commonly used in triathlons and time trial events on road and track. However, they are illegal in most mass start road races or any other event where drafting is permitted because, while aerodynamically advantageous, they tend to draw the hands away from brakes, make the rider slightly more unstable on the bike, and can be dangerous in the event of an accident. Further, they are not useful in sprints or shorter climbs where power is of greater importance than aerodynamics.

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