I've already shared my Rule of Thirds tip (link to that post). As I implied in that one, there's more to composition than getting something "in the crosshairs." Here are a few things that can help every photo you take be just a little more interesting.
Step in closer--one of the biggest things people tell me they don't like about their pictures is that after a few years it can be hard to tell who's who. They're talking about the big, group shots we're all tempted to take at family reunions, etc. Instead of that, try getting smaller groups and don't stand 15 feet away so you can get their shoes--get in close. Get just their heads, or head and shoulders. If you've been watching television or magazines lately you will notice that it's not always necessary to take the whole face. I actually don't recommend this technique unless you have lots of time to practice it but it demonstrates that recording the outfit people are wearing and the shoes they have on generally isn't necessary.
Diagonals--add interest to your photo by having a clear diagonal for the eye to follow. I have no idea why this works--I'm sure it has something to do with psychology or something, but if you are photographing a path through the woods, see if you can stand in a location where the path will move from top right to bottom left (or the other way around). As you think about this, you'll find lots of opportunities to let this work for you--an exterior staircase with a rail is something I like. If it fits what you're doing, go for it. A children's slide in a park--rides at a carnival--even the diagonal piece of lumber that helps hold the barn door together--all of these are diagonals that can work for you.
Odd numbers--it's a "rule" in many areas of our lives. To add interest to a group of pictures on the wall--use an odd number of them. When arranging flowers, if you have something you really want to stand out, use one, three, etc., as long as it's an odd number. On a house, an even number of windows at the front of the house works because there is a door in the middle to make the arrangement have an odd number of "entrances." Take advantage of this when taking photos.
Uncluttered backgrounds--if you want someone or something to really stand out, keep the background as uncluttered as possible. I don't mean they should stand against a white wall, but to have one person standing in front of a crowd of 100 isn't going to make the one person stand out. If you're outdoors, the picture on the beach or the grass of the park, even in front of a fountain (as long as the fountain's water doesn't appear to be sprouting from your subject's head) are all just fine. Sometimes you can't pull it off--don't worry, get what you can and keep it in mind for the next time.
There are always going to be exceptions to every "rule." Don't feel you have to all of this every time. Perhaps you want a photo of your daughter dressed in her ballet costume and you're in a room full of audience members--no problem. Your daughter will still stand out, even though the background is full of people. Always remember, the first rule of photography is to have fun.