Ignoring the elephant.
You have to make a conscious effort to see everything that's in the frame, not just the subject that you're so interested in. Classics include ignoring the drain-pipe growing out of your subject's head on the wall behind them.
Small red objects in a predominantly green or blue picture can throw your carefully composed arrangement, because red tends to draw the eye strongly. I've taken loads of pictures of a field full of classic cars that are ruined by some prat in a bright red coat.
Having more than one subject is another good one; having a person in the foreground with a really interesting background doesn't make for a good picture. In fact, that's a good point. You have to decide if you're taking a snap for the family records, in which case you're usually talking person plus situation. Or you're taking a picture, in which case composition is the most important thing.
Tips about composition: Try to position your subject on what are called the major thirds. If you took a ruler and drew lines down and across the picture at one thirds intervals, where the lines intersect is a major third. An object on one of these points should jump out.
In a landscape, don't have the horizon half-way up the picture. Either the main subject is the land or its the sky, so have more of the main one.
The above should have so tied you up in knots that you'll never take another picture, you'll be so busy composing and worrying about the exposure and so on.
It takes time for your brain to develop a 'picture mode' where everything you see is processed to see if its a good picture. Then you'll realise how many great pictures you'll see when you don't have your camera!
