I've been playing with my DSLR for months now without understanding what shutter speeds and apertures really do. After doing a bunch of reading on the science of exposure and the intuition of exposure, I finally get it. At least I think I do and now I'm going to embarass myself by revealing my ignorance. Please tell me if I get something really wrong.
A photograph is properly exposed if the camera has the right aperture, shutter speed, and ISO film/sensor sensitivity to capture most of the light coming in from the scene. The actual science is complicated (the magic word is EV), but the way photographers talk about it is in terms of "stops". If a scene is twice as bright then you need one less stop in the camera to make your photo. Conversely, opening up a stop doubles the amount of light recorded.
Shutter speed is one way to adjust stops; if you double the length of the exposure you double the amount of light captured. Common hand-held exposures in one stop increments are 1/30s, 1/60s, 1/125s, 1/250s, 1/500s. My Canon 350d lets me adjust shutter in 1/3 stop increments such as 1/30s, 1/40s, 1/50s, 1/60s. So I have to click the wheel three times to double the amount of light recorded.
Aperture (or, confusingly, "f-stop") is another way to adjust exposure. Opening the lens aperture one stop will double the amount of light coming in. The numbering is a bit confusing because of the geometry; doubling the f-stop number will cut the amount of light coming in by one quarter, two stops. Common one stop increments for aperture are f2, f2.8, f4, f5.6, f8, f11, f16. My Canon 350d lets me adjust aperture in 1/3 stop increments such as f2.8, f3.2, f3.5, f4.0. So just like when adjusting shutter speed, I have to click the wheel three times to double the amount of light recorded.
Finally, the third way to adjust exposure in a camera is to change the ISO sensitivity of the digital sensor (or film). Doubling the ISO will make the sensor twice as sensitive to light, which is about the same as doubling the amount of light recorded. Common ISOs in digital cameras are 100, 200, 400, 800, 1600. My 350d only lets me adjust ISO in full stop increments (not 1/3), so every time I "click the wheel" (or push the button) I'm doubling the amount of light recorded.
That's a lot of detail, but what it boils down to is:
A modern camera with light metering means you can mostly be ignorant of this stuff, just put the camera in P mode and click the wheel to trade off shutter speed and aperture for artistic effect. But now I understand just what I get by bumping up one ISO level: I can cut the exposure time in half. And this information is a foundation for more complex topics. The next thing I'm trying to understand is dynamic range; just how wide a range of EV various scenes have, how many EV levels my camera can capture in a single photo, and how many EV I can print or view on a screen. This article is a good start.