This is an edited version of what I posted as a comment on Sarah's blog.
Photography is essentially an art form. Composition makes photographs interesting. To improve your art skills, find photos you like and study them, asking yourself: 'Why exactly do I like this picture?'
When you take a photograph, to identify what the subject, ask yourself: 'What is the purpose of this photograph?' and 'What is the reaction I want a viewer to have?'
Next find a 'context' -- a simple backdrop which adds relevance, contrast, and/or location to the 'subject.' You can add depth by finding a 'context' in a different spatial plane than the 'subject.' For example, if the subject is a building in the background, make the context a flower or person in the foreground.
I think a great photograph is a subject, a context, and nothing else. Remove any clutter that detracts from your message. Get closer -- zoom in -- and crop as tightly as possible. Take photos at various angles and distances to give you more options to choose from. Also experiment with vertical and horizontal camera orientation -- you may discover that this simple shift yields interesting results.
The one big difference I've seen between an amateur photographer and a professional is this: the amateur will take one picture and move on while the pro will take many pictures of the same scene at varying angles, distances and exposures before moving on.
Remember that the center of the frame is the weakest place -- it's static, dull, and gives no value to the context. The more you move the subject away from the center, the more relevance you give to the context; so juggle until you get the right balance.
Use the "Rule of Thirds." Imagine a tic-tac-toe grid over the image you want to take. Use the intersection of those lines to place your subject slightly off center, and bring interest up or down. Horizontal lines are peaceful; diagonals are dynamic or tense; and curves are active and sensuous.
Dawn and dusk provide lovely light, whereas midday light can be harsh. Take advantage of overcast days for photography. That's when the light is even, and casts almost no shadows. Inside, avoid using your flash. Instead, provide as much natural light as possible -- open curtains, or even the door.
Where do you go to find the most interesting images? Do you have a favorite place or do you simply like to look around and see what you find? When you are driving, look for:
Light (shadows and highlights)
Shapes (round and angular)
Color (harmony and discord)
Texture (rough and smooth)
Composition (strong and weak)
Tones (light and dark)
Patterns (even and odd)
A picture is a playground for the eyes to explore, so provide a path of movement, and some space for the eye to rest. Your personality and your vision must come through in every photo you take.