Keep it simple
Think to yourself, "What am I taking a picture of?" and keep that in mind. Identifying the subject of interest and avoiding distracting backgrounds will help to keep the picture clear. Zoom in to clear out irrelevant parts of the scene and capture just what you're looking for, avoiding objects like signs, buildings or people that take the viewer's eye away from the point of focus. An example of this is taking a picture of crowd of protestors - a busy image where the eye has trouble figuring out what should take its focus. Zooming in on one protestor in particular, though, makes it very clear what should command the viewer's attention.
Rule of thirds
Picture a tic tac toe board: two horizontal lines intersected by two vertical lines. This creates an easy formula - line up the horizon of the shot with either of the two horizontal lines, and line up the subject (either a person, building or the focus of your picture) with either of the vertical lines, ideally where the lines intersect. When viewing a scene, try to overlay this map into the viewfinder - with only a little adjustment, you can quickly create more visually interesting images by simply adjusting (or cropping after the fact) what you see to line up with these invisible markers. When dealing with a moving subject or a person, it's often preferable to have them looking or moving 'into' the picture from one of the two sides.
Lines and shapes
We all remember our geometry classes, dominated by circles, triangles, and snake-like curves. Applying these simple shapes to your subject matter can help to simplify complex scenes and add visual interest. Consider trying to capture an image of a person walking down a long, straight street. Instead of shooting straight down the line, move yourself five or ten feet to the side and shoot that road at an angle - having that line crossing through the intersecting lines of the imaginary tic tac toe board from the rule of thirds can create the illusion of movement as they lead the eye through the picture. S-curves are even more dynamic, while repetitive lines can also create movement of the eye through the picture, like repeating waves of sand on a beach or parallel row houses along the side of a road.
Most images taken by amateur photographers are taken at eye level - this means most of these pictures are taken from the narrow range of 5 to 6 feet in height. Taking a picture from a lower vantage point (for example crouching or even lying on the ground) can add grandeur and significance to the subject, while getting more height (from climbing up a tree, fence or steps) will reduce the significance of the subject in your scene. Examples of using this could be taking a picture of your children playing looking from the ground, or capturing a busy marketplace scene where no one person would stand out over another.
When considering what you're capturing, look through the lens and pick out the dominant subjects, like people, buildings, trees or mountains and arrange them so that they compliment each other. This can mean either symmetrical balancing, where objects of equal size are positioned on either side of the picture's center, like a manicured garden with bushes on either side, or asymmetrical balancing, where objects of different sizes are used on either side of the picture's center, like a scene of a person standing between a house and a tree. Asymmetrical pictures are often more interesting and visually stimulating as the viewer's eye moves from object to object.
Framing, as it sounds, is a way of drawing attention to the subject in the picture by blocking off or framing parts of the scene using natural or artificial barriers, and however accomplished can add prominence to the subject, and will help add a sense of depth to the photo. Using this concept literally, you can try taking an outdoor scene from the inside through an open window to create interest, or capture a newly married couple kissing in a doorway or hallway to draw the eye to them. Other more natural ways of framing a shot are using trees (shooting through gaps in the branches and leaves), or viewing a beach from between craggy rocks.