Simplicity, Simplicity, Simplicity!
What makes a great photograph? That's a hard question to answer, but simplifying your composition is unquestionably one of the most powerful approaches. Here are some simple ideas to help in your quest for simplicity.
Find What Really Interests You
The best place to start is with what you like. Something made you stop and get your camera out in the first place. Don't leave that as a vague feeling but give it some real thought. If you can ascertain what you find appealing in a scene, you can create a more compelling image by focusing on just that. Pay attention to what moves you and use it as your guide.
Get Close to Fill the Frame
Once you determine what interests you, zoom in on it, or get physically closer to it so it fills more of the frame. This will have the side effect of cropping other objects out and thereby reducing distractions. But don't fall victim to bulls-eyeing your subject. If you find it useful, make use of the rule of thirds to position your subject on a power point within the frame. If the edges of your subject sit too close to the edges of the frame, it will look crowded. Either leave it a little room to feel more natural, or get even closer to eliminate the edges of your subject entirely, relying on just the key features of the central portion to convey your intent to the viewer.
Eliminate the Horizon
Getting rid of the horizon avoids the duality created by cutting the frame in half. Don't leave the viewer wondering which side of the line to focus on. Unless what draws you to your subject is the horizon itself, get rid of it. You don't need it.
Use a Shallow Depth of Field
The depth of field you select through your choice of aperture and working distance can have a dramatic effect on what a viewer sees in your image, and what you have hidden from them by rendering it out of focus. With this, you can isolate just the narrowest slice of your subject so as to show enough to convey your intent without cluttering the frame with more than is necessary.
Reduce What You See to Graphic Shapes
Once you start to get into the swing of things, try looking at your subject not as what it actually is, but merely as a collection of geometric, graphic shapes. A pleasing arrangement of simple shapes will automatically result in a simplified composition. Don't be afraid to go completely abstract either. Even if the resulting image ends up unrecognizable in terms of real world objects, graphic simplicity can still lead to great photography.
Periodically Reevaluate Your Subject
Just because your subject started being one thing doesn't mean you can change as you go. As you crop out everything but your subject, and begin to crop out the less interesting parts of what remains, you may find that your attention is now drawn to one particular component of your original subject. Don't worry that you can no longer see everything you started with as this is part of simplifying. Don't be afraid to go with it. If you want an insurance policy, snap a few extra frames as you go.
Employ These Guidelines More Than Once
As you get closer to your subject, keep looking to home in more closely on what interests you about it as well. Don't let just your initial assessment be the end of your explorations. As you work, take a moment now and then to appreciate what you've found and reevaluate what draws you to it. This sort of iterative approach allows you to comfortably settle in and become more intimate with your work.
You Can't (Or Shouldn't Assume You Can) Fix it Later
You can do a lot with digital editing, but some things are better done in your camera, in the field. Attempting to blur the background digitally will rarely come out looking as natural as would true lens blur from shooting at the proper aperture. Cropping digitally can cut out surrounding distractions, but at the cost of losing a good portion of the megapixels you paid for when you bought your camera. And regardless, one of the benefits of simplifying your composition is the enjoyment of the process itself. Discovering new ideas and new images as you work can be a lot of fun. Why would you want to try short circuiting that process with some mouse clicks on your computer after the fact?
Henry David Thoreau famously laid down the dictum of "Simplicity, simplicity, simplicity!" in his book Walden. Although he clearly wasn't writing about photography and composition, his advice in this expression is nonetheless quite applicable to our attempts to create great photography. Take your time, and keep these three words in mind as you work. You may surprise yourself what you end up with.