CORE CONCEPT - A SIMPLE COMPOSITION RULE 
The principles of composition can be distilled into one sentence: Amounts and distances should always vary somewhat. This is a confusing phrase, not always easy to understand at first. Even if you do understand its implications, it is not always the case, and it IS possible to make a strong artwork that violates this in some way, but generally this rule will make an art piece stand on its own better. To clarify how this works in practice, let's look at a few examples. 

LIGHTNESS - If we divide brightness into light, dark, and midtones, there should be a different amount of each. Say, 20% dark, 35% midtones, and 45% light. If the amounts of each are the same, this is complete symmetry and is boring. If it's entirely dark or light or midtone, that's even worse as no contrast exists at all. 
 
HUE - Likewise, warm, neutral, and cool colors should vary in amount. No two of these should cover equal amounts of the artwork surface. Note that the area around an artwork can be cosidered in some cases - contrast with the wall the artwork is mounted on can allow less contrast in the work itself. For example - a work that is overwhelmingly warm and neutral with few or no cool tones may stand out against a cool-colored surface. Context does have an effect on the perception of an image. 
 
SATURATION - vivid, moderately vivid, and dull colors again follow the same principle. No two amounts the same. 
 
FOCAL POINTS - These are usually important objects in the image you want the viewer to notice. People and peoples' faces inherently draw attention, as do objects that contrast with the majority of the image in some way. Lines converging towards the object (formed by the scene around it) will also direct attention towards the focal points. How, you're probably asking, do focal points follow the one-sentence rule? They do - in their placement. The RULE OF THIRDS is a well-worn concept in art, in which a canvas is divided into a 3x3 grid, and the focal point is placed on or near one of the four intersections. The rule of thirds is really a shortcut to follow the sentence described above; the grid forms intersections at places that are not in the center or the edge of the canvas, either horizontally or vertically. In other words, you're positioning focal points such that in both axes, the distances to the edge of the canvas vary. They aren't the same, and they're not drastically different either. (After all, placing a focal point such as a face at the edge or dead center of an image is visually awkward. If you've ever seen a horribly framed photo with a face poking out of the bottom of frame, or the top part of the face extending out of the top of the frame, you know what I mean.) Note further that most images will contain a mix of multiple focal points, and all of them should follow this rule. 
 
BALANCE OF MULTIPLE SIMILAR FOCAL POINTS - We've stated that distances should vary somewhat, not drastically. This means that there's a sweet spot or at least a range of positions, somewhere between total symmetry and total asymmetry, which often is argued to be centered around the golden ratio. The golden ratio, which is defined as half of one plus the square root of five, or approximately 1:1.62, appears throughout nature, and seems to be somehow subconsciously appealing to the human mind. Most photographs, sheets of paper, electronic screens, and art canvases are not too far off from this height/width ratio. Often it makes sense to break up areas of similar brightness, hue, saturation, etc, such that the 'big bright red object' (or equivalent item that stands out) off on one side of the canvas is conterbalanced by a smaller or several much smaller similar patches of a similar color on the other. These counterbalances are usually of similar but not the same size or intensity. In other words, the relative strength of multiple focal points also generally adheres to the rule outlined above. 
 
BONUS RULE, AKA THE SECOND IMPORTANT SENTENCE: 
 
DRAW THE VIEWER INTO YOUR ARTWORK. 
 
What this means is that focal points, lines and shapes should provoke the viewer to look at your work for a while instead of directing them to look elsewhere. Often this involves a web or circle [loop] of notable lines and shapes, which lead the viewer's gaze around from one point of interest within the artwork to the next, in a cycle of some sort. For example, a portrait of a woman gazing at a flower vase in front of an arched window, might cause the viewer to glance initially at the woman's face, then follow the direction of her gaze to the flowers in front of her, then following the cloth of a curtain along the curve of the arched window, back to the woman again. The effective use of composition can hold attention for a few seconds longer than would otherwise be the case - and will make your work more memorable as a result.

You can see some of my attempts to use these principles in my own artwork [on Etsy, Pinterest, etc] through the below links.
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That said, I'll openly disclose that when you buy something from another vendor using these links, for example, a link to Amazon.com, I will get a small fraction of the sale amount and will use that as a means of supporting and expanding the projects I'm working on here. These ads for other vendors, as well as - especially! - links to my own shops, are helping me keep these sites online and growing with new content on a more frequent basis.