Composition can generally be defined as the relative placement of subjects and objects in a piece of work. Composition is extraordinarily important. In many sense it is the key measure of the artistic value of a photograph. While you may be able to learn the technical side of photography quickly, it might take a lot longer to master composition. There is no limit to where one can go with it. Composition is often what elevates photography to an art form.

The goal of composition, if there is a goal, is to draw a viewer's attention to the most significant part of your photo. Sometimes that is by using focus, color, contrast, exposure, juxtaposition, geometry and so on.

An implicit goal of composition is also to develop a personal style. A statement that highlights your sensibilities or value judgement.

While there are probably infinite ways to understand composition, it is helpful to know about some common styles that often help define compositional elements.

Patterns. Geometric patterns such as circles, squares, rectangles or arcs that repeat in a photo. Patterns may also be philosophical and abstract.
Symmetry. Clear symmetry and clear lack of symmetry make interesting statements
Texture. When light hits objects at odd, low angles, interesting textures emerge in surfaces. Food photography can rely on textures to create a sense of realism in food.
Depth of field. Perhaps the most obvious and overused. However, there is something ethereal about clever use of shallow depth-of-field images.
Lines. Another common style is to use various types of lines (diagonal, horizontal, vertical, converging, diverging) to draw attention to or to lead the viewer toward or away from a specific point in the image.
Rule of thirds. Keeping your subject just off-center - 2/3 from the left (or right) and 2/3 from the bottom (or top) - can make for very compelling images.
White Space. Empty space plays an important role in composition. Too much or too little can both be used for interesting effects.
Framing. Natural framing is a common way to make a composition more interesting.

Kids protecting themselves from a sudden downpour

Tips for making good compositions

While there are no rules for making great compositions, there are certainly guidelines for making good ones. As with most other things - once you have mastered the guidelines - you can break all the norms and create your own masterpieces.

Keep the line of horizon straight.
Avoid placing subjects bang in the center
Use rule of thirds - put subjects just right of center and above middle
Use lines to draw attention to the subject
Use physical and philosophical repetitions and patterns
Avoid shooting words along with other subjects
Take close-ups for more dramatic effects
Remove everything that isn't adding to your composition