The Rule Of Thirds
This is the most fundamental rule of image making there is.
First, let's look at some examples of photos that use the rule of thirds. Check out the following site (occasionally updated by yours truly) for a whole pile of example photos that follow this compositional guideline.
What Is The Rule Of Thirds?
Fold a piece of paper into thirds. Do it again the other way. You know have a tic-tac-toe board of sorts, on the image. The lines and intersectiosn made by the creases, those are the third lines.
When humans look at rectangles, we tend to look at these points and lines first. We also tend to observe things that fall on these lines more than things that don't.
This image (shot on an Olympus OM-1) puts the ladder on the left third line.
How To Use The Rule Of Thirds
Thus, when we are trying to figure out where in the frame to put something, the simple rule is this: Put it on the third. On an intersection or line.
Landscape? Don't put the horizon line in the center, this is uncomfortable. Move it to the top or botton third line!
This image puts the horizon line, blurry as it is, on the bottom third. The subject is, of course, on the left; and their face on the top-left intersection.
Taking a close up photo of a face? Too big to put it on a third? Put the eyes on the top third. Done, now worry about something else, like the expression or the lighting.
The Rule Of Thirds is Everywhere
This rule is considered by many to be the most fundamental guideline (concrete, actionable guideline) when taking a photos. When you crop a photo in most software, thirds lines appear to help you decide. Most cameras allow you to put third lines over your viewfinder/preview to see while taking pictures.
In this photo, the subjets are on the bottom-right third line, the road on the bottom line, and the building horizon is about the top third line. I didn't think about any of this when I took this photo, I just saw the evening light providing a rim light around my friends, a nice scene of a city; I dropped back a few steps, pulled my Olympus XA out of my pocket, and snapped the photo.
Break This Rule
Of course, with such a fundamental rule, it will get broken. All the time. I encourage you to break it! Break it on purpose, as an exercise in understanding other compositional elements, as they must be present to "take over" from this rule, which can largely be a starting point.
The easiest example is scenes with symmetry, or stronger compositional elements that matter more - like leading lines, or the photographer's physical restrictions (I couldn't go there, or get close enough, etc).
In this photo, the leading lines mattered more.
The second reason to break it is, by breaking it, you can draw a lot of attention to the object that "doesn't fit". Big pictures of walls with a subject in the corner, for example.