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For a good depth of field example, check out this image on Dooce's blog. Note that the battery on top of the nose is in sharp focus, the one at the back of the head isn't, and the wall is pleasantly blurred.
In photography, depth of field (DOF) refers to the range of distances from the camera that appear to be in focus. A lens can only focus at one distance. However, the sharpness as one moves closer or farther away from that distance diminishes gradually and within a certain range nobody notices it.
Perhaps you want to take a photo of a friend standing in front of another object. If you have a narrow (or short) DOF, your friend may be in focus while the foreground and background appear out of focus. At the other extreme, a wide (or long) DOF could result in the entire image appearing to be in focus.
There's a good example of effective DOF control here.
We've all taken photos of someone outside or in front of a window, only to find out, later, that the person's face came out so dark that we can't see it. To sum it up, the issue comes down to this: Some more-sophisticated cameras can detect and compensate for backlighting. But, if the camera thinks that the backlight is lit 'normally', it will underexpose the rest.
Entire books have been written on the subject of composition and it's difficult to do the topic any justice in a single post. However, for those of you who are just starting out, I'll offer two suggestions: Fill the image with your subject and consider the 'rule of thirds'.
It is virtually impossible to buy a digital camera without automatic exposure capability, so it's no surprise that most digital camera owners don't learn the very basics. Good automatic exposure systems result in decent results most of the time, but as you have likely noticed, in others the results are disappointing. If you want to avoid photos that suck, you need to understand three basic exposure controls: Media sensitivity, shutter speed and lens aperture.