(other photogs, if you think I worded something wrong, by all means let me know so I can fix it!)
So let's dive in!
Okay, the first thing you need to learn is what a "stop" is. A stop, or f/stop, is a... 'unit' of light. It is technically a function of the relationship of the opening of the aperture and the focal length of the lens. That's as scientific as I will go for a definition. At its basic level, it is a measure of light. If you double the amount of light in an exposure, you are increasing the exposure by "a stop" or "one stop" or "one f/stop." If you halve (as in, divide by two) the amount of light in an exposure, you are decreasing the exposure by a stop. So you see, you just need to understand that it's an exponential scale, doubling light as you increase, halving it as you decrease (the amount of light).
Where cameras today complicate things is that many allow you to increase or decrease by half or third stops. The principles are the same, but avoid those half and third stops for now. I will list the standard full stop increments for aperture, shutter, and ISO:
(your lens' "iris", the opening that allows light in)
<< opening up wider, more light <--- f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22 ---> closing down, less light >>
Some lenses, particularly expensive ones, will let you open up even wider, like f/1.8, f/1.4, or f/1.2. Some also let you close down more to smaller apertures like f/32, f/45, or f/64.
The smaller the number, the more light comes in to hit your sensor, because it's a fraction, see? You know that 1/2 is bigger than 1/4, right? That's how you remember. So if you are in a low-light situation, you want to "open up" your aperture to let the most light in as you can. On cheaper lenses, you may be limited to just f/3.5 on the open end.
Outside where it's bright, you'll want to "stop down" or "close" your aperture to something like f/16 or f/22, to make sure you don't overexpose. You see, since there is more light, you can use a smaller opening, just like a small fraction is 1/22.
Aperture also affects depth of field. The wider open your aperture is, the 'shallower' your depth of field is. This means less of the subject will be 'in focus' in the photo at f/2, than at f/8. This is a common effect in close up portraits or macro shots. Deep depth of field is achieved by "stopping down" or "closing" your aperture to a smaller value like f/22. This is useful for shots where you want most everything in focus, like a landscape or architecture shot.
Depth of field should only be considered after you make sure you have enough light to "play with." If your situation has low light, you are stuck with wide apertures and thus shallow depth of field, because if you "stopped down" to f/8 or f/11, it would be too dark.
Shutter speed is how quickly the shutter covering your film or sensor closes. The most important rule to remember is that for hand-held shots you will want to shoot at AT LEAST 1/60 (shutter speeds are in fractions of a second) or faster. People with steady hands can go all the way down to 1/30 (which is a slower speed), but it is risky. I have heard a general rule is to shoot no slower than 1/(focal length), so for a 200mm telephoto lens, you may want to shoot no slower than 1/200th, and for a wider lens like a 35mm, you can shoot slower at 1/30th.
Just like apertures values, cameras nowadays come with half and third stops in shutter speeds, too. I will list the standard 'full stop' shutter speeds below. Remember, each of these is either half or double the amount of light of the speeds beside it, depending on which way you go. Like 1/30 is double the light of 1/60, and 1/125 is half the light of 1/60. See? Fractions (of a second).
<< more light, slower shutter <-- 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000 --> less light, faster shutter >>
Some cameras (most today) will allow slower shutter speeds slower than 1/2 a second (many allow for 30 seconds), and also for faster than 1/1000 of a second (mine will go to 1/8000). The effect of shutter speed is on motion captured. Slower shutter speeds will alow you to capture fluid motion, such as wheels spinning, water flowing, car lights trailing, etc. Slower moving subjects require slower shutter speeds to show their motion, like maybe a two second exposure of a waterfall to have it nicely smoothed out. On the flip side, fast shutter speeds like 1/250 or 1/500 can freeze actions such as human movements like jumping or running. These will have no blur. Even faster speeds like 1/1000 or 1/2000 of a second can capture water splashing or something along those lines.
As with aperture, these motion effects can only be considered once you know there is enough light to use. If you are indoors, you are not going to be able to freeze fast motion if you have to shoot at 1/30s. Just like if you are out on a sunny day, you won't be able to get motion trails from someone running since you'll have to shoot a faster shutter speed to keep from over exposing the photo.
ISO is related directly to what used to be called ASA, or film speed. ISO stands for International Standards Organization. But that's totally irrelevant.
What you need to know are the standard ISO values, and that just like aperture and shutter, they either double, or halve the amount of light in a photo. It is a function of the sensor itself, telling it how sensitive to be. There are also third and half-stop values for ISO, but like I said, you need to know the standard values. They're easier than aperture and shutter.
<< less light sensitivity, less noise <--- ISO 100, ISO 200, ISO 400, ISO 800, ISO 1600 ---> more light sensitivity, more noise >>
So, you ever see color specs all over your photos? That is from shooting at a high ISO. In the film days, high ASA film would have lots of larger silver crystals in it and that's what you know as "film grain." High ASA film produces grainy photos, whereas high ISO sensor settings produce "noisy" photos. There is absolutely no reason at all to shoot at a higher ISO than you need to. Where there were effects gained from shooting at different shutter speeds and apertures, with ISO, the image can only get worse and degrade as you increase your ISO. Not only will you get bad noise, but you will start to lose highlight and shadow detail. What you have to figure out is what is the acceptable amount of noise in your photos, in your opinion.
Different cameras can handle noise better than others. When I shot with my Rebel XT (Canon 350D), I would never shoot anything serious over ISO 400. With my newer Canon 5D, I can shoot up to ISO 800 or higher sometimes, and not worry about it. But the 5D is one of the best cameras for dealing with noise, so I would try to stay at or below ISO 400 for most purposes. And like I said, ONLY GO HIGHER IF YOU NEED IT. If you are outside, shoot at ISO 100, or if you camera has ISO 50, use that.
You get the idea? I know that was a really brief part but if you have questions, let me know.
So, let's look at some examples of how to use all this info.
Okay, let's say you're going to be shooting outside today. It's sunny out, so you're going to go to a wildlife park or something. (come on, don't laugh! stick with me). You put your camera's ISO on 100 (of course, since you have plenty of light, right?), and you use Auto mode to get a meter reading. It says to shoot at 1/250th at f/11. Well, you see some birds flying around, and you want to "freeze" them in the air. 1/250th just isn't cutting it, the shutter needs to be faster because they're getting blurry. So, you want to up your shutter to 1/1000th of a second. Now switch to manual mode, set your aperture to f/11 like your meter said, and set your shutter to 1/1000th like you decided. Shoot. Uh oh, it's way too dark. That's because using aperture and shutter and ISO is a give and take situation. Since you gave more speed to your shutter, you went from 1/250 to 1/500 then to 1/1000. That's two stops of light you lost. Where do you get it back? Well, keep your ISO at 100 for now, since you're outside. All that's left is your aperture. So if now your picture is 2 stops too dark, you need to OPEN your aperture 2 stops wider. So 1 stop wider puts you at f/8. 2 stops puts you at f/5.6. So set your aperture to f/5.6, shutter to 1/1000, leave ISO at 100. Take the shot. It's spot on, not dark anymore! See, a 1/250 @ f/11 is the same as 1/1000 @ f/5.6. You just shifted your values down the line. Get it?
So if you want to slow your shutter way down to get a nice, smooth waterfall in that creek over there in the shade, what will you do? Well, there's a little less light here in the shade, so you up your ISO to 200 (that's not too high, it'll be fine). Okay, take a meter reading. 1/60 @ f/5.6, ISO is 200. Well, you want to slow your shutter speed to 1/2 a second to get that waterfall smooth and not "frozen" in time. Okay, so count how many stops difference that is. 1/60 to 1/30 is one stop, 1/60 to 1/15 is two stops, 1/8 is three stops, 1/60 to 1/4 is four stops, 1/60 to 1/2 is FIVE stops! Okay. So you're going to be overexposing your picture by FIVE stops right now if you just leave everything the same and change your shutter to 1/2. Five stops would be adding 2 (double), 4 (double again), 8 (double again), 16 (double again), 32(double again)-- 32 times too much light! That's going to turn out solid white if you shot it right now. So obviously, we need to lose some of that light! How? Start with your aperture. You are at f/5.6 right now. You need LESS light, so we need to CLOSE the aperture five stops down. So, one stop is f/8, two stops is f/11, three stops is f/16, four stops is f/22. Uh oh. Your lens only closes to f/22. That means you'll still be over exposing by 1 stop, which means it will be twice as bright as it needs to be.
BUT WAIT! You upped your ISO to 200 when you went into the shade, didn't you? Well, if you take that back down to ISO 100, that will give you your fifth stop of light.
So now you see that 1/60 @ f/5.6, ISO 200 is the same as 1/2 @ f/22, ISO 100. The light level will be the same. But how will the images look different? Well, since you closed your aperture from f/5.6 all the way top f/22, you will have really deep depth of field. Nearly everything will be in focus now. At f/5.6, you might have started losing focus in the background. Okay, shutter speed. That was the whole reason to change everything, right? At 1/60th, the water would have been pretty much frozen still. But now at a full half a second (1/2) shutter speed, the water has moved enough to blur smoothly. Last, you took your ISO down from 200 to 100, so you end up with less noise!
Keep in mind, that since you slowed your shutter from 1/60 to 1/2, you would obviously need to set your camera on a tripod or rock or something, because you are NOT going to be able to hand-hold a shot with a shutter of 1/2. That's too long, by a wide margin.
Hope all this helps! I'd post more examples, but I'm at work and so yeah, I should probably get back to work. Let me know if any situations arise, and I'd be glad to help. Just drop me a note or something!