Ok...so here it is. I make no claim to be an expert on the subject. This is just based on what I've learned from reading and my own experience. If anyone sees errors or disagrees with anything, speak up! I wrote this quickly. Again, this is a basic guide for those that know little or nothing about basic exposure concepts. Of course there are exceptions to everything. Sorry if my explanations aren't as clear as I think they are.
Exposure Essentials 101
The key to a good picture is proper "exposure", how well lit the picture is. Of course, the lighting is controlled by factors such as light sources and shadows, but inside the camera there are three things which will control the exposure:
Shutter speed
Aperture size
ISO sensitivity
The first two control how much light will strike the image sensor, and ISO determines how sensitive the sensor is (think of it like the different film "speeds" back in film photography...ASA 100, 200, 400, etc.).
All three of these interact with each other. Think of it like a basic algebra problem:
x= proper exposure - this is a constant, it never changes
a= shutter speed
b= aperture size
c= ISO
So, x = a + b + c. Okay...there really isn't any kind of addition going on. BUT, there is a relationship between everything. If you change one value (a, b, or c), you MUST change at least one of the other values to compensate...if not both values. To further illustrate, I'll just use some random numbers. If:
10 = 5 + 2 + 3
If I change 5 to 6, then I MUST change one or both of the other values to maintain the outcome of 10:
10 = 6 + 2 + 2
or
10 = 6 + 1 + 3
See? It's about relationships, and how they all interact with each other.
You will have different "exposure modes" on your camera (not including the auto or "scene" modes):
**Aperture priority - You set the aperture (and ISO), and the camera will determine the shutter speed for correct exposure.
**Shutter priority - You set the shutter speed (and ISO), and the camera will determine the aperture for correct exposure.
**Manual - you set aperture, shutter speed, and ISO. Correct exposure is up to you to figure out.
Each has uses and advantages. You will probably find that aperture priority will be the most useful in most situations. This is why most photographers, even the pros, use this most of the time. This is because the aperture size controls the "depth of field", and this is something you usually will want control of (I'll discus "depth of field" in the next section). But there will be times when it won't give you the results you need, and you will need to pick another mode...depending on what you're trying to achieve.
Now, to explain what aperture, shutter speed, and ISO does:
** Shutter speed - controls how fast the shutter stays open. A longer shutter speed will let more light in, making a brighter picture. But for a picture of a moving object, a faster shutter speed will be needed to freeze the object. A faster speed will let LESS light in, making a darker picture.
** Aperture size - the aperture is the opening that lets the light in. The values used for this are a little confusing. It is expressed in "f-stops". A smaller f-stop value means a larger opening, and a larger f-stop value means a smaller opening. So, an f-stop value of f/ 2.8 is a very large opening...letting in a lot of light. A value of f/ 16 is a very small opening, letting in much less light.
** ISO - as stated above, the ISO is the sensitivity of the image sensor. A lower number means it is less sensitive, and requires more light to strike the sensor for the image to be bright.
What are the uses of these settings? I'm glad you asked.
** Shutter speed - you can use this to either freeze a moving object...using a fast speed. Or, you can show movement of an object by using a longer speed...and letting the image blur as it moves. Also, you can use a LONG shutter speed on a non-moving object to let in much more light. Such as...taking a picture of a landscape...at night. You will need a very long shutter speed to let in enough light.
** Aperture - besides controlling the amount of light, it also controls the "depth of field". That is...how much of the image is in focus...from the front of the image to the background of the image. If you use a very large opening (like f/ 2.8) only a very small portion of the image will be in focus...the area you choose. The rest of the image will start to blur, very softly. This is used in macro or close-up photography, but also used a lot in portraits...where you WANT the background to blur, so it won't distract you from the main subject.
Just the oppposite...using a small opening (such as f/ 16) will make most of the image in focus, not just the main subject. VERY useful for landscape photography, where you want the WHOLE image in focus...not just some little part of it.
** ISO - you're thinking...
if a higher number means the sensor is more sensitive...then I'll just use a high setting...right? WRONG.

The higher the number is, the more noise or "grain" will be generated in the image. In general, TRY to use ISO 100 when possible. It will provide the cleanest image. However, sometimes it just isn't possible to use ISO 100, because you just don't have enough light.
So, now for some examples, to maybe make this make a little more sense:
Example 1 - Outdoor portrait, aperture priority:
I want to take a picture of you outdoors, where it is very bright. I want you to be in focus, but I want the bacground to blur and be soft. So, maybe I'll use an aperture of f/ 2.8. I'll leave my ISO at 100. Since it's very bright outside, this will be probably be fine. In bright sunlight, shooting at ISO 100 and f/ 2.8...I'll probably have a very fast shutter speed. Since you will be standing still, this isn't important. But, if you were moving, this would be a nice benefit.
Example 2 - Nature landscape, aperture priority:
I want to take a picture of the pretty landscape. It's a bright, sunny day. I'll leave my ISO at 100 to start with. Since I want the entire image in focus, I'll use an aperture of maybe f/ 16. This will probably give me an acceptable shutter speed, enough to hand hold the shot. If not, I will either need to put the camera on a tripod...or increase the ISO, which in turn will make the shutter speed faster. Sense I'm increasing the sensitivity to light, the shutter speed will get faster to compensate for the extra brightness.
Example 3 - Action photograph, aperture priority:
I want to take a picture of my child, while he's running, in bright light. But, I also want to control the depth of field for a more pleasing image. I'll use an aperture of f/ 2.8...a large size, letting in a lot of light...but also nicely blurring the background. But since he is moving, I need a fast shutter speed. At ISO 100, I may not have a fast enough shutter speed to "freeze" the action. So, if I start bumping up the ISO...maybe to ISO 400...or even ISO 800...I'll get a VERY fast shutter speed to compensate for all that extra brightness. If the depth of field isn't important to you, you could set your ISO somewhere in the middle, maybe ISO 400, and use Shutter priority. Then you can set exactly the shutter speed you want...maybe 1/1000 sec. And the camera will determine the proper aperture setting. (but, it may not look the way you want...so just be aware)
Example 4 - Night time landscape of the city:
You can try aperture priority on this, but you may have to switch to manual and just try a few settings until you get it right. But try aperture priority first. The same as the above landscape...I want everything in focus. So, an aperture of f/ 16. But this is a low-light shot...I want the ISO to be very low, so I will not see noise in the image. So, I'll use ISO 100. Because of the small aperture and low ISO, I will need a LONG shutter speed. Maybe 30 seconds....maybe more. This is why you might have to switch to manual mode...to get longer than 30 seconds. A TRIPOD IS REQUIRED FOR THIS SHOT.
A last thought: if your camera lens doesn't have "image stabilization", you will have a hard time taking hand-held shots at shutter speeds slower than 1/60 second. You may be able to, depending on how steady you are. If it does have "IS", you can MAYBE use slower speeds...maybe down to 1/20 sec. This all varies from person to person. If your subject is moving, you will need to use a faster shutter speed (maybe 1/100 sec) and possibly a flash. Image stabilization won't help you with this.
Flash photography gets a little more complicated, and some of these rules don't apply in the exact same way. Get comfortable with this before attempting any serious flash photography. Walk before you run.