I've got a few requests from family and friends to help them better understand their photography, and to explain a few basic concepts, which is what I will attempt to do here.
Essentially, without going into anything terribly fancy like studio strobes, or flash or off camera lighting, I'll boil down the basics of photography to three important factors: Shutter Speed, Aperture (also known as F-stop) and ISO (also known as ASA). I'll attempt to use some of my own photographs to explain these basic concepts.
Let's start with shutter speed. Essentially, the shutter is like the "eyelid" of an human eye, except it spends the majority of its time "shut." When the shutter opens, it lets in light, and exposes your image on your sensor (or film), and stops the exposure when it closes. However, you can control how fast or how slow your shutter opens and closes, thereby determining how much light is impacting your image.
For example, let's use your own eye as a model. Close your eye for a second, and then blink normally and close it again - what did you see? You probably only saw enough for one quick look at something, and have an image in your brain and froze whatever action was occuring at that moment you blinked. Now, let's try this a little bit differently. Close your eyes again, and blink once more - however - keep your eyes open for about 2 seconds rather than a quick blink. What did you see? I suspect you saw a lot more going on during that two seconds and a lot more light to see whatever it was you saw.
Here are a few illustrative examples:
Shot at 1/125th of a second - about the same time as an eyeblink:

See how the battery is frozen while it was rolling down the tray? It doesn't look like it's rolling, because I used a "fast" shutter speed to freeze that movement.
Now, the next image should prove something interesting - Shot at a full second - not very long by our standards, but literally an eternity in photographic terms considering how fast light moves.

As you can see, the battery itself is blurred out, because it was moving during that 1 second time exposure and as such, was blurred. One second doesn't seem like a very long time, but for something that's moving fairly quickly, it's actually very long. Additionally, see how the edges of the frame are blurred also? I'm holding the camera, and therefore I shake a little bit in that one second - that causes blurriness too!
Moving onto the next concept - Aperture, also called F-stop. While shutter speed is the time that you leave your shutter open, Aperture is the "hole" that allows light to pass through to expose the image. Think of aperture as your pupils of your eyes - when it's bright outside, your eyes automatically reduce the size of your pupils to better manage the light outside. When it's dark, like in a movie theater, your eyes make your pupils open up more to let more light in so you can see better. This is a very important concept to understand, because you're not going to be able to shoot at the same aperture every single time - depending on how bright/dark it is, you might have to make your aperture smaller/bigger.
Now, I should probably explain aperture numbers - you'll see numbers like F22, F11, F8, F5.6, and you probably don't really understand what they mean. Here's the rule of the thumb - the bigger the number, the smaller the hole. In other words, Numbers like F32 implies a very tiny aperture, and as such, probably won't let in that much light. Numbers like F2.8 implies a BIG aperture, and as such, will let in a lot of light.
Now, when do you determine when to use such apertures? When you have a lot of light to work with, generally smaller apertures are ok. If you're in a dark area, and not very much light is available to use, a bigger aperture (like F2.8) is the way to go. Additionally, smaller apertures do not allow you to keep everything in focus - a small aperture will generally keep one thing in focus while throwing out the rest into a blur. Great for portraits. However, you'll most likely need to increase your ISO depending on how much light you have to work with. Large apertures will show a lot of detail, and keep a lot of things in focus - great for landscapes. However, you'll most likely need to use a tripod, because the smaller apertures don't let in as much light as a larger one, and as such, will require more shutter speed time to expose properly.
Now, onto the LAST concept, which is ISO. ISO is essentially the sensitivity of the sensor to light - the higher the number, the MORE sensitive it is to light, and vice versa. In other words, a lower ISO will probably require MORE light to expose properly as opposed to using an higher ISO, which requires LESS light to expose properly.
Most entry level DSLRs have ISO ranges from ISO 100 to 1600, and the more fancy pro level cameras can go up to as high as 24,600 ISO, some even more. You might be asking yourself - why not just shoot at the highest ISO possible, to get the best amount of sensitivity? Noise. The higher you go up, the more noise or "grain" you will introduce into your image, and thus degrade the quality of the image.
Now, you're probably getting how all of this works - now how does all of this come together? One thing these three concepts have in common is that they ALL control light in some fashion. Shutter speed controls light via time, Aperture controls light via openings, and ISO controls light via sensitivity. To gain in one area, you need to "trade off" in another. For example, to get a faster shutter speed in a lower light situation, I need to increase my ISO so I can increase my shutter speed - the trade off is the image quality of using a higher ISO. Learning how to balance all three is the key to exposing an image properly, and as such you need to ask yourself these questions and evaluate your situation:
Shutter speed: Is my subject moving? Do I need to stop the action, or create a blur? If these are critical components, then shutter speed is important to you.
Aperture - is my background distracting? Do I need separation between my subject and the background? Am I shooting a landscape where I need a lot of detail? If this is the case, you need to decide if a small/large aperture is what you need.
ISO - the last factor - how much quality do you want? Can you live with a little bit of noise, or do you prefer to have none? 100 ISO is no noise, 1600 is going to be very grainy.
This concludes my basic tutorial.