Perspective
Photographing your subject straight-on is sometimes the right choice, but you can create visual impact by moving the camera left, right, above, and below. When you’re beneath the subject it often makes them/it appear more powerful to the viewer. Conversely, when you’re above the subject it makes them/it appear more diminutive. You can use this to an extreme for a powerful impact, but it’s also a very good subtle technique for portraits. Slight positioning above or below the subject can subconsciously imply aggressiveness and passivity (respectively) without being too, uh, obvious…
Additionally, left and right positioning isn’t as direct and can often make a photograph feel more honest and candid. When capturing a moment, whether it’s staged or not, photographing the subject head-on can often seem a little awkward and end up being less-effective.
Of course, you can also combine different positioning elements to create other effects. Try taking photographs of the same subject from different perspectives and see how people interpret them. This is a good way to understand the effects your choices have on the end result.
Use Shapes and Lines to Draw the Eye to a Specific Point
The viewer’s eye doesn’t magically end up looking at one of the intersections in the rule of thirds grid, it’s just more natural. That said, if you have a reason to draw the eye elsewhere you can accomplish that pretty easily by choosing where you place shapes and lines in your photograph. A shape doesn’t mean a literal, detail-less shape, but in the sense that a building could serve as a rectangle. Roads often make nice lines in landscapes. When you’re composing your photograph, consider the shapes and lines and where they draw your eye. If they’re taking you out of the photograph or away from the primary subject, you’ll probably want to consider a different composition. Let the roads lead where you want the eye to go.
Perspective can even make a road straight ahead appear like a triangle and draw the eye into the horizon. Whatever the case may be, make sure your shapes and lines are taking the viewer where you want them to go.