I say I am an intuitive photographer. Intuition allows me to quickly and intuitively compose within a frame. I don't think I always have had this skill. It developed over time. I always have looked at a lot of visual media: art, paintings, photos, comics. Ever since I was little, I have enjoyed going to art museums and spending time looking at other peoples compositions. As a kid I drew a lot. In retrospect I believe that if you look at how masters compose their art, and try to emulate their compositional techniques, eventually some of that amazingness will rub off on you.
Not everyone has an art museum nearby, but there are other ways to be exposed to great visual composition within a frame. Spending time with one particular book really has made a difference for me. That book is Sister Wendy's 1000 Masterpieces, by Sister Wendy Beckett. You can find it used on Amazon for under 10 dollars. Pick up a copy and spend time with it. Just look and absorb how the great masters composed their paintings. Maybe try to apply what you pick up to your photography. Sister Wendy's commentary is well worth reading as well.
Where do you start with photographic composition? Presumably you are now walking around with a camera in your pocket. Preferably your camera is a phone, or small and inexpensive “Point and Shoot”. You are always carrying it. You are identifying things that might make good subjects. When you find a subject, look at it with your camera.
Your camera provides a frame. See in terms of that frame. In your mind, a frame can visually be divided into sections. Start by imagining 4 sections. Is your subject balanced – is it symmetrical or asymmetrical to the frame. Notice you can move your camera viewpoint to change this balance. Even moving a little ways right or left, up or down can drastically change how an image looks in a frame. Try an experiment where you take a number of pictures of the same subject from different vantage points. When you review them later, are there vantage points that you prefer? Identify these and reuse them in future compositions. Increase the number of sectors you visually divide your frame into. Try six.
Your camera very likely will have the ability to zoom from a very wide view to a very detailed narrow view. This seems to often be the first step in a photographers composition process, as most cameras make zooming in and out very easy. Your choice of zoom level will very much depend on the subject and what surrounds it. A long and wide view establishes place. A medium view evokes daily life similar to what the eye sees. A close-up view emphasizes detail and intimacy. An ultra close-up macro view shows unique details that the eye would normally never catch. Ask yourself how distance from your subject affects the visual story you are telling. At this point don't worry about technical details like the focal length of your lens. That can come later.
Try and determine the path of the eye in the frame that you see. There will usually be something obvious that draws the eye, and usually that is the subject. Then there will be something secondary to the subject that my eye sweeps to, which might be secondary to the subject, and so on. The eye tends towards an “S” path sweep of a scene. Move your eye through the frame like that. Try reversing the “S”. Move the camera right, left, up or down to improve the path your eye takes. Notice how subtle changes in camera position change the path of your eye. The camera can be tilted to change viewing angle. Tilting your camera can simulate motion and make your image seem more dynamic.
Your composition has depth – some objects will be close to the camera, some will be of intermediate distance, and some will be far away. There will be a foreground, a middle-ground, and a background and a perspective relationship between the three. Foreground will tend to be sharp and backgrounds tend to fade. Where is your subject in terms of the depth of the photo? Wherever it is, you want it to be in focus. This may mean that other parts of your composition are less in focus. You can control this, it's called depth of field, but I don't want to get overly technical here so won't go into that. Often the middle-ground is the best place for a subject. Move about to achieve that. There are exceptions – a sweeping landscape for example. Often prominent objects in the foreground and background, if they are not the subject, can be very distracting in a composition, so try and minimize those. Carefully placed, a less important object in the fore or background can be complementary to the subject.
Objects in your composition will have relationships to other objects within the frame. Position and placement in relationship to other objects affects impact. Objects can overlap or not, and how they do might or might not be pleasing. Objects will relate to each other by relative size. You've probably seen the image of the far away horse and the close up hand that appears to be holding the horse, and that is a trick of manipulating the distance and position of objects.
You will see areas of light and dark. There are colors that compliment and oppose. Good compositions can emphasize differences between light and dark colors, or present very similar colors. Complimentary colors, which are colors that look good together, come in pairs – black and white, red and green, purple and yellow, blue and orange. Each color has shades – dark green, middle green, light green. Fall: September and October is the perfect time to experiment with complimentary color combinations, as the landscape becomes so colorful with the change of season. Winter is starker, tending towards browns, grays and yellows. These colors are less complimentary and more similar. Shape often takes precedence over color in winter.
Get a quality lightweight tripod. One that you might carry in a small day pack. A tripod allows you to compose deliberately. A tripod lets you to make and lock in subtle changes in viewpoint. Almost all “Point and Shoot” cameras can be attached to tripods. Phones will need a special device that will allow attachment of a phone to a tripod. You don't have to use it all the time, but set aside some time occasionally to tripod photography and see you your compositions improve.
The best thing that I think you can do to improve you composition is to spend time looking at how great artists, painters especially, frame and compose their work. And then try and apply these ideas to your own compositions. Composition is something you can really practice without too much effort and get noticeably better at!