The basics of photography are a simple equation made to appear complicated by using obscure notation.
Exposure = Shutter Speed x Aperture x Sensitivity
Exposure is the total amount of light captured, and the camera attempts to guess at what the correct exposure is. When it's guess is not what you had in mind you use Exposure Compensation to tell it what you want, or go to full manual mode and take control, though the camera will still show you how far you are away from what it thinks.
The reason why you can change different parameters is that in addition to affecting exposure they also have additional effects on the image.
Sensitivity is the simplest since it is just the gain applied. The tradeoff for increasing the gain is more noise and therefore less image quality, but otherwise doesn't change the image.
Shutter Speed affects how motion is captured, including your own motion while holding the camera.
Aperture is the most complicated. First it's expressed in an obscure form such as f/2.8 that you can't plug into our equation, but just know that one click of the dial when you change the aperture has the same effect as one click on the dial when you control the shutter as far as the exposure is concerned. What it affects in the image is called the Depth of Field, which is how far in front and behind the plane of focus will be sharp. But Depth of Field is also affected by the distance to the focal plane and the focal length of the lens. This is where there is no substitute for experience. The smaller the aperture (bigger f number) the more depth of field.
One of the main reasons for using a dslr is that the image sensor is larger, which means you use longer actual focal length lenses than a compact camera. This in turn gives you less depth of field for the equivalent field of view. A compact camera with a small sensor gives you very little control over the depth of field except at very close focus distances.
That's it. The various modes of the camera are just about which of the variables you are going to control and which you are going to let the camera control. But only in Manual mode will the camera not balance the exposure out for you, so don't be afraid to turn the dial away from Auto or Program. I use Aperture Priority 90% of the time, since that's the one that has the most effect on the image.