My first post was a laundry list of compositional elements. However, each element's usefulness varied. Some that I've seen repeatedly amount to little more than superstition. The utility of others depends on context. So by itself, the list's utility is limited.
So what things lead and please the eye? What things help with understanding? The most common ones might be:
Contrast: (in any sense, for example, size, colour, or shape)
Leading lines
Faces/text
Lines, shapes, forms, patterns, textures, symmetry and colours all have varying influences. These elements are significant only if none of the above are dominant.
Many other things can influence the effectiveness of composition. My only claim is that from my point of view, the three above are the most common and so need to be front of mind. Other influences come from Gestalt theory.
Figure-ground: Viewers want to separate images into figure and ground.
Similarity: Similar things are seen as part of the same group, creating harmony.
Proximity: Close objects are seen as a unit. Stronger than similarity.
Closure: The mind tries to complete a shape or pattern.
Continuation: A series of like objects seen as a line. The line will extend beyond.
Symmetry and balance: Symmetry = stability, Asymmetry = dynamic, engaging.
Isolation: An item that is isolated stands out
Common Fate: Items that move in the same direction are perceived as related.
Law of emergence: The whole thing is identified before the parts.
Simplicity: Easy to understand is better