The Rule of Thirds:
When taking a photo, imagine it being evenly divided into a grid of 9 squares (3 across and 3 down).
Generally, you want your subject to be in the areas where the lines intersect, because it makes for a more visually interesting photo.
When you place your subject off center, like in the rule of thirds, you might leave a void in the scene which can make it feel unbalanced. You should balance the "weight" of your subject by including another object of lesser importance to fill the space.
When we look at a photo our eye is naturally drawn along lines. By thinking about how you place lines in your composition, you can affect the way the image is viewed, pulling us into the picture, towards the subject, or on a journey "through" the scene. Pay attention to the lines created by your surroundings (horizon, roads, trees, buildings and structures, and even sunlight) and use them to help frame your subject.
Cropping and Framing:
When deciding how to crop your picture, you want to eliminate distracting background elements whenever you can. You'll want to decide how important your subject is and then give it a ratio of the frame that is directly related to its importance. Often the main subject becomes lost among the clutter of its surroundings. By cropping tight around the subject you are elimating distracting backgrounds, ensuring the subject doesn't have to compete for attention.
The same is true of framing, you'll want to frame your subject in such a way that the surroundings help draw the viewer's attention toward the subject. Often it is appropriate to take a shot that puts a person in context with their environment. Just be mindful that the surroundings don't take attention away from your subject and add unnecessary visual clutter to an image when cropping and framing.
Perspective is how your subject is portrayed in your image relative to it's size or position within the frame. The easiest way to change the perspective is to move and change the viewpoint, or the angle at which you are shooting. For example, shooting an object from high up can make it feel small, where as shooting from below will give the subject a sense of dominance. You can also achieve this by shooting from a distance or close up. No matter how you shoot it, perspective can dramatically change the mood of a composition.
The best rule of thumb when it comes to choosing a background is to ask yourself if it will add to your composition by adding context, or will it be distracting and leave your subject competing for attention? The best rule is to keep it simple and free of distracting elements. When we look at a scene, our eyes naturally gravitate towards points of interest while ignoring unimportant information in the surroundings. The camera does not discriminate in this way, it will simply capture everything in front of it, so it is important to be mindful of your background. You want the surroundings to draw your eyes to your focal point, not leave it competing for attention.
Photographs are a two dimentional representation of a 3-D image. The camera has a tendency to flatten the depth in an image, so it's important add dimension to your image. You can separate your subject by blurring the background, using a wide aperture. To keep the background in focus, use a smaller one. You can include something in the foreground, middle or background, which helps the eye separate them as different "layers" which it interprets as depth.