What's the difference between a good photograph and a bad photograph ? Assuming you've gotten the colours right at the printers, you don't have a thumb in the way, and you have proper focus and exposure, photographers are always talking about what are known as the "rules of composition" when discussing the technical merits of a photograph. Depending on who you talk to, there are great differences in opinion about how many actual "rules" there are. I was taught that there are 14. And for no other reason than that, that is what I'm going to teach you.
1) Center of interest
Every photograph has one of these. What's the image about? Is it a picture of your wife? Is it a picture of the Great Pyramids of Giza? It should be very obvious with a single glance to every viewer what the photograph is "about". While it's called the "center" of interest, it isn't necessarily true that the subject of interest is in the center, or takes up a large part of the image, it is nevertheless very obvious that that is what the photograph is about.
This is the first rule of good composition, because it is the most important. Decide before hand what it is that you are taking a picture of. Everything else you do is then focus around
bringing that center of interest out.

The center of interest is obviously the violin player. Not the wall, not the violin, etc.
2) Fill the frame
This rule is here for mostly practical reasons. Film costs money. Even digital images will cost money to store and print. Once you decide what your center of interest is, fill the view finder with it. Get as much detail as you can. However there is a point at which it becomes too much. You want to fill the frame but you also want to leave some background and foreground to give a sense of context to your subject. I personally use around 80% as a guideline for how much to fill. That's not a hard and fast rule and I break it all the time myself. It's really up to your good taste and the effect you want to create.

There's a whole building and square behind and around this statue. But I chose to only photograph the statue and filled my frame with it.
3) Lines
Lines are extremely important in visual arts. Lines give us shapes and contours. Lines are what lead the viewer's eyes from one part of the photograph to another.

The zig-zag lines and twirl at the bottom of the stairs gives a modern twist to this.
4) Flow
After just finishing lines, we should talk about flow. Flow is how the viewer's eye is lead from one part of the photograph to another. One way to give flow to a photograph is to use lines. They can be horizontal, vertical, diagonal, converging or diverging lines. Sometimes the flow created by lines is sharp and distinct (such as the sides of a building converging up into the sky), or they may be less obvious (such as a line of irregularly spaced stones that mark out a farmer's field). Nevertheless, the viewer's eye should be able to spot the same elements from one part of the image to another. Flow creates the illusion of motion (or lack of motion if so desired). Diagonal, converging or diverging lines are considered to be "dynamic" while horizontal and vertical lines are considered to be "static". A careful balance of dynamic and static elements will give an overall sense of motion to your photographs.

The motion blur and leading lines gives the viewer a sense that everyone is headed into a distant point.
5) Direction
Direction is similar to flow. It also creates the illusion of motion. If something in the photograph appears to be "moving" then it most likely has a direction in which it is moving. One example of what I mean is the little stick man that you find in many traffic lights. One stick man seems to be standing up straight, legs together, arms next to the body. Most people will agree that he doesn't appear, visually, to be moving. This is obviously the "do not walk" signal. His counterpart has bent arms and bent & spread legs. He appears to be moving. This is obviously the "safe to walk" signal. The "safe to walk" signal has a direction in which he is moving as indicated by the bent arms and legs. The "do not walk" signal seems static and has no direction of motion. Direction in photographs can be created in a number of ways. A lone figure in the middle of a convergent path appears to have direction despite the fact that his arms and legs may not be moving. That's because we feel as if we were to see him a split second later, we just know, he'll have his legs moving down the path. Similarly, a partially captured car just entering the frame from the left will lead us to think that in the next split second it will be at the right of the frame.

The line of figures gives the viewer a sense that everyone was walking towards me. Which infact, they were.
6) Repetition
Repetition of some subject (a bunch of hot air ballons, a flock of flamingos) gives a sense of togetherness to parts of the image. For instance, a group of hot air balloons may all be heading off in one direction. That group may create interesting patterns in the sky and add direction to the image. It can also have important psychological factors such as giving a sense of togetherness and friendship.

The repetition of the trees gives a sense of solidity. This photograph actually works because there's essentially two different levels of contrasting brightness.
7) Colours
There are warm colours and cold colours:
warm - red, orange, yellow
cold - blue, green, purple
There's a whole psychology thing with colours. Red's are "firey". Blue's are "tranquil". I won't go into it too much since I know there've been entire books written on the subject. So it should be fairly obvious why this is an important element of composition.
With colours, pay attention to contrast as well. Contrast is defined as the difference in the level of brightness between the brightest and darkest parts of your image. However, contrast is also created by placing warm and cold colours next to each other.

The bright yellow center is surrounded by a field of cooler green.
8) Groups of 3's
I've been told there's something psychologically appealing about groups of 3's. One is lonely, two is too well balanced and static, four and above is too much of a crowd. But 3 is just perfect !!
You know, as in Goldielocks and the 3 bears, 3 Stooges, 3 Musketeers, 3 colours on a flag, books come in trilogies, etc. Some how we humans just like 3's. Don't ask me why, I'm not a psychologist.
Back to photography. If you are going to photograph a group of something, either go big and photograph a bunch of them, or look for groupings of 3's as your center of interest. It works, trust me !

What a gorgeous group of ladies !
9) Rule of 3rds
This has been around for a long time. If you go into a museum and look at any really good painting. Then imagine a 3x3 grid on top of the picture, you'll notice that important elements such as eyes, windows, tops of mountains, horizon lines, etc. all line up with either one of the grid lines or at one of the intersections of the grid lines. It's claimed that by having your center of interest at one of these points or lines, you get a much more interesting image.
Note: some cameras allow you to super-impose such a grid pattern on the view finder to aide in composition. Check your manual.

Notice the position of the chandalier. Notice the position of the box. They're exactly on the "power points" according to the rule of thirds.
10) Negative Space
Basically large white or black space in the background (usually off to one side, but I've seen it done in other ways as well). Psychologically, we humans want to fill that space with our imagination (kind of like a cartoon thought balloon). It's a great trick because you can't go wrong. Whatever fills the negative space is interesting to the viewer because he/she came up with it herself !!

The butterfly is nicely highlighted by the white sand.
11) Foreground/Background
What's in the foreground and background is obviously important. This includes elements from the other rules, such as bright colours, or converging lines. But sometimes you can have too much in the foreground or background which will distract your viewer from your center of interest.

My little ducky travel companion forms my foreground, while the location is obviously placed by the background.
12) Framing
Look for things such as doorways, bridges, signs and other things to frame your subject under. It gives a sense of perspective and intimacy to the image. Ever notice they always have the bride walk through the big door into the church ? Ever notice how long hair can be used to frame the features of a face ?

Framing need not be created by solid frames. I positioned the camera so that the trees framed my subject.
13) Clutter
Clutter is having a lot of "stuff" in either the foreground or background. All those different shapes and shadows can distract (or in some cases if done well add to) from the center of interest. Cluttered images seem chaotic and disorganized. This is not necessarily a bad thing and may be used to convey a sense of dynamicism.

The band and equipment forms a very cluttered group.
14) S-Curves
A lot of people love S curves. It's very sensuous and sexy. It's the shape of a woman's profile. It's also a meandering river, or slithering serpent. It's very dynamic, but not a fast, hasty kind of motion, but a slower more sensuous movement. It's also harder to find in nature, so is considered more interesting.

The dancer's body forms a natural and energetic S-Curve.
So there you are, the 14 "rules of composition". How you use them is going to be up to you and your good taste.