Before you just step up and take a picture you should consider what you want your viewers to look at and how you should display main points of interest. You should ask yourself, what is the main subject? What angle should the light be hitting in my picture? Is there anything that could accentuate the main subject? Where should the main subject be in the frame? These are all important things you should consider, but that doesn't necessarily mean you need to follow the rules exactly.
The Rule of Thirds has been used for centuries and is probably the most important of all the composition techniques. The Rule of Thirds means that the frame can be divided into three horizontal sections and three vertical sections and therefore, where the horizontal and vertical lines intersect makes an ideal location for the more important parts of your picture. By locating your main subject at one of the four intersections you give the subject more emphasis than if it was right smack in the middle of the picture. This is also a good technique if you have more than one important subject, the intersections can still work even if there's a subject on more than one. The divisions can also be helpful in setting up a picture, they can for example, help you determine how much horizon you want. Most famous photographs or paintings in the world today have the rule of thirds applied to them in some way.
Simplicity is the method of keeping the information in a photograph relatively simple. If your main subject is close, then your background should be very simple to avoid distractions. You should try to keep everything not important much less interesting than what's important in the frame. Especially avoid lines or objects that lead the eye away from the subject.
Framing is the tactic of using natural surroundings to add more meaning to your subject. It could be anything such as bushes, trees, a window, or even a doorway like in the picture at the top of this page. In the process of doing this you need to be careful that you don't only focus on what's framing your subject. Make sure you focus on the main subject, and also it is a good idea to use a narrow aperture (high f/stop) to achieve a high depth-of-field. It also wouldn't hurt if the part of the picture framing the subject was darker so make sure you take your light reading on the main subject.
Texture can add a significant amount of interest in any picture. When people see texture in pictures they start imagining what it feels like to touch what's in the picture. Texture is a good idea when your taking pictures of rocks, walls, surfaces, someone's hands, or leaves. In order to make a picture reveal a texture you must make sure the light is coming almost exactly from the side of the surface so it creates shadows in places key places.
Leading Lines are used to lure the eye deeper into a picture or to an important subject. Straight, curved, parallel, or diagonal lines are all good at promoting interest. Good examples could be roads, rivers, streams, bridges, branches, or fences but there are endless things that could be used.
Colors are what add heart and emotion to your pictures. Certain color configurations can inspire awe and amazement in onlookers. Colors can be used to add all sorts of accents and effects, but you must be careful to not draw attention away from the main subject.
It might not be a bad idea to keep these key terms with you when you practice taking pictures. The best way to learn and improve your composition is just lots of practice and experimenting.