It is more important to think about what to leave out than what to actually capture.
I have found that in every good photograph, whether technically or artistically, there is some truth to this. What is not shown draws more attention to what is shown, and what out of focus draws more attention to what is in focus. I will use some examples of mine to illustrate what I mean, but I do not claim to be a good photographer by any means.
1. Composition - Photo Link
When choosing what to leave out, the first step is to consider composition, i.e. what is in the frame and how things are oriented and positioned. In this first image there's a lot of negative space at the top of the photo. This empty space forces the viewer's eye to the lower half of the image, where it is darker. There is little else in the photo than can distract the viewer—no clouds, no boats, and in fact, no color.
2. Bokeh - Photo Link
Bokeh (bo-kay) is a Japanese term that refers to the "blurriness" of the background (optimally achieved at lower aperture with a shallow depth of field). With the bokeh blurring the background, the focus of the image is thrown on the actual subject. What is left out of this image is this blurred background. Photographers pay for SLRs and expensive lenses for this kind of effect. Bokeh makes the subject pop a whole lot more than if everything is in focus (unfortunately endemic to all point-and-shoot cameras by virtue of being compact).
3. Macro/micro - Photo Link
This is the photography of things very close or very small. Because of this, the background is typically blurred as a result of getting very close to the subject and the background is relatively further away. Most cameras have a macro mode (usually with the image of a flower). Of course, it is not as good as a dedicated macro lens. Consider a macro lens as being the microscope used to see the invisibly small. Because of the enhanced bokeh effect with macro photography, I would argue that it isn't too difficult to get a good macro shot no matter what the subject is. Still, it is an art and world unto itself.
4. Background - Photo Link
A simple background simply gives whatever is in front of it more emphasis. It is essentially left out of the picture, as it were.
5. Lighting - Photo Link
With lighting and shadows, it is all about what is included and left out. The word "photography" essentially means "light-recording". It is about how light is captured and how shadows, i.e. the lack of light, gives a subject texture and depth. In this photo, the subject is clear and it is the way the light affects the rest of the image that is interesting.
I am a firm believer that it is more important to think about what to leave out than what to actually capture, merely as a means to emphasize the subject of the photo. It doesn't actually matter how that is achieved. I just keep this guideline in the back of my head when I do go shooting. Let me know how it works for you.