Like most things in life, there are rules and guidelines used in order obtain the best results. Same goes for photography. You may have heard of the "rule of thirds" before. It is an invisible grid used to compose a picture. In nature, odd numbers are most pleasing to the eye. (If you've ever looked at an arrangement of flowers, chances are it's an odd number.) By applying this "odd number approach" to your photographs you can achieve an image that is visually pleasing. It all comes down to imagining a grid across your frame. (Some cameras and phones have the option to show a grid on your screen.) Where the lines intersect is where you'll want to place your object. Take below's picture for example; while out sledding with my nieces and nephews I turned and happened to see this scene.
It is a perfect example of utilizing the rule of thirds! Just imagine their heads coinciding with invisible intersecting points. To make it easy, I plopped a grid over the picture below.
By having my subjects off to the side, it creates a more interesting focal point. Allowing my viewers' eye to wander around the whole image instead of staying in the same spot.
When using the rule of thirds on portraits, it's important to remember to to face your subject looking "in" towards the picture. We, as humans, are naturally drawn to eyes. In turn, we look to see where the person is looking. If I were to have my brother facing left instead of right, the viewer would fail to acknowledge the rest of the photograph since their gaze would be following Kenny's off the page.
The rule of thirds is also great for creating negative space. Negative space is "empty" so to speak, like a sky or ocean, or anything that is a "solid" compared to the rest of the image. In the above landscape, the sky is the negative space, leaving the tree line and mountain to follow the rule of thirds.
The rule of thirds is also excellent at directing your viewer where to look - especially in crowded scenes. In the image above there is a lot going on. But your eye is instinctively drawn to my niece and from there your eye branches out and absorbs the rest of the image. By framing my brother's profile on the edge it stops your eye from wandering off and instead directs it back to the middle of the image.
One of the key reasons to know the rule of thirds is knowing when it's ok to break that rule. That's half the battle! I break the rule of thirds all the time, but I do it intentionally. The above two pictures would not have the same effect if they were composed off to the side. In this case, the snow being negative space would have distracted the viewer and allowed eyes to wander.