Rule of Thirds

The "rule of thirds" is one of the most basic concepts of composition.  Draw imaginary lines through an image dividing it into thirds both horizontally and vertically.  This gives nine small rectangles the same ratio as the original image and four points of intersection between the horizontal and vertical lines.  Each of these four intersections is a "point of interest" where the eye of the viewer will naturally be drawn.  An object of interest placed one of the lines will be perceived as more prominent in the image.  An object of interest placed at an intersecting "point of interest" will be perceived as much more prominent.

Subject(s)

Placement of one or more elements of the subject at (or very near) the intersecting points adds power to them and makes them much more noticeable.  Where the subject(s) are placed is important.  If there is a direction of view or feeling of motion, the subject needs to be looking or moving into the photograph.  In the case of the fisherman photograph, his placement at the lower right intersection captures not only him but a good portion of the the view he has of the lake.  In the photograph with the two lovers looking at each other, their heads are at the two upper intersecting points.  The viewer is immediately drawn to their faces and the fact they are "making eyes" at each other.  There is another compositional element working with the rule of thirds in both of these photographs called "hidden leading lines" and are the directions the subjects are looking.  There is more about "leading lines" and how they work in that section.

Horizon

A horizon should normally be placed along (or near) the upper or lower of the two horizontal lines.  Without some compelling reason for doing so (see the exceptions), placement dead center splits the image unnaturally.  If severe enough it can seem like two separate images, an upper and lower.  Which line to use depends on the scene, what is to be depicted, and whether the sky or the foreground is of greater interest in portraying the subject.  In the case of sunrises or sunsets, the horizon is usually placed along the lower horizontal line.  In other photographs, the location of the subject and distant hills or mountains may dictate the location of where the sky meets them.  Uneven horizon can also affect placement requiring thought about balancing how it appears in the image.  Exact placement is not essential, only approximate, and how well it fits with the intended subject and how it is to be portrayed is an equally important consideration.

Lines

There is more detail on "leading lines," perspective and vanishing points under its own topic.  Under this one, however, use of straight and curved lines along (or near) "rule of thirds" lines, or to connect their intersecting points tend to be aesthetically pleasing.  Connecting or intersecting where a "rule of thirds" line intersects the edge of the image also works.  Note also the line need not reach an intersection point, only head or "lead" to it.  It's not as powerful as actually crossing it, but the eye and brain will normally extend perspective lines beyond where they end in the image.  There is no magic formula for how to align strong straight and curved lines in an image with the "rule of thirds" lines and their intersections (including at the image edges).  Very rare is a scene that has everything perfectly placed.  When composing an image, practice at identifying the strong lines that are present and try different perspectives placing them at various points.  Just as exposure can be bracketed, how various elements and prominent lines in the photographs align themselves with the "rule of thirds" lines and intersections can be "bracketed" by taking photographs from multiple perspectives, sometimes just slightly different.  If necessary (and possible), move a little closer or a little farther.  With enough conscious practice, the hunt for aligning elements and lines will start to become automatic, and when it appears in the viewfinder the brain will lock in on it as simply looking "right."  Grid lines are not required in the viewfinder for exacting placement; approximate works.

Exceptions

What would a rule be without exceptions?  There are a couple of notable ones to the "rule of thirds."  The most often encountered are exceptionally strong symmetry for architectural and strong reflection for scenic (and sometimes windows, chrome finishes, etc.) which is another form of symmetry.  Very strong symmetry in an image will override nearly all other compositional factors.  In the several images provided as examples of exceptions, there may be some modest use of "rule of thirds" lines and intersections.  Most commonly discarded is locating the horizon either high or low and placing it dead center.  The nature of the symmetric subject will determine its placement in the image.  In the case of the hood ornament it felt more natural to have it leaping into the image versus leaping toward the edge.  In the case of the pond, the strength of the water reflection and its general shape allows a centered horizon to work with a few other subject elements intersecting the image edge near a "rule of thirds" line.  If in doubt, hunt for a pleasing subject placement in the viewfinder and "bracket" the perspective taking several photographs (if possible) trying different horizon locations and positions of very strong lines.  In general though, it is best to let a strong symmetry express itself with the balance of the symmetrical parts matching the balance of how they are placed in the image.