All images © 2015 Steve Miller
This topic relates mostly to DSLRs and digital still cameras with Manual exposure modes. We typically recommend avoiding scene modes because the camera doesn't necessarily understand what you might be trying to accomplish (ever try to take a photo in AUTO in a bar?). However, an understanding of the camera's aperture can still help you manipulate your results even when shooting in automatic scene modes.
Our primary method of shooting wide angle involves fixing the camera's ISO and shutter speed, then varying the aperture to balance the available light. Whatever is in the foreground will be automatically properly exposed using your strobes' TTL exposure (as long as you're close enough).
Here are 3 different treatments of the same type of subject, same time of day—only the Aperture has changed. None are wrong or right... just different exposures. When you think of aperture, think of the water column (available or natural light) behind your subject. A larger aperture (smaller f/number) will provide a lighter background, while a smaller aperture (larger f/number) will provide a darker blue background.
In some cases, a smaller aperture improves the edge sharpness of wide angle photos. This is particularly useful to know when shooting with an external wide angle wet lens. A photo taken at f/14 will have sharper edges than a photo taken at f/4.
The traditional macro approach is to "stop down" the aperture (increase in f/number) as much as possible to increase the depth of field, which is to say the range of distances which is in focus. Typical apertures for macro photography are between f/16 and f/22. A small aperture lets very little light into the camera, so your strobes will need to be powerful and close to your subject.
A small aperture is also used to create those stunning shots of small critters on soft corals with black backgrounds. The key is to choose a subject with a fair distance between it and the coral in the background. Otherwise your strobes will illuminate the coral in the background and cause it to show up in your image.
Don't have your camera handy? We love the Camera Sim App which allows you to virtually tweak camera settings and see their effects on a photo. Try varying the aperture while leaving the rest of the settings alone.