They are used when composing your shots,
They make your photo shots more visually appealing,
They draw the viewer to a focal point in your photo, making your shots more interesting and unique
Probably one of the most important rules of composition is the 'rule of thirds', meaning the 'focal point' of your shot should rest along one of the grid lines or where they intersect each other, as shown in this infographic.
The reasoning for the 'rule of thirds', is that it draws the viewer's eye to a point in the photo. If there is no focal point as such, then the viewer is left to try and find the focal point of the photo, or the photo may have too much detail; that no focal point gets established at all. Using the 'rule of thirds' helps to draw the viewer to a specific place in the photo.
For this ocean landscape photo, there is not a 'focal point' in the 1st photo, because no obvious focal point is at the grid lines or intersects. We see the sky, the ocean, the sand, but our first look at the photo does not draw us to any particular 'focal point'.
However, if we change the photo so that the horizon in the photo lies along the top grid line, then we can say that the photo is now composed as per the 'rule of thirds', and although we still see the sky, water, and sand, the photo now draws more attention to the reflection on the water, the focal point in the photo.
In the cropped photo - 3rd photo - the sky is now 1/3 of the photo(background) , while the ocean and sand is the 2/3 of the photo (foreground). You also could have the sky as your background at 2/3, and then the foreground in your photo would be 1/3 -the water and sand.
It really depends on your photo, if the sky is really great with nice colors, then you may want to showcase that by making it 2/3 of your background, or if the foreground is more appealing, you may rather give it more weight in the photo. This is the essence of the rule of thirds. If you have a landmark, a statue or people in your photo as the 'focal point' the same rules apply, photograph them along the lines as shown.
To help practice the rule of thirds, especially for novice photographers, most cameras have a 'grid line', that can be enabled in the camera settings, which shows while taking your photo.
Another Example - Rule of Thirds:
This photo of Peggys Cove is not composed per the rule of thirds. We want the focal point to be the lighthouse and the sunset over the water. So we will crop it so that the lighthouse will be along the lower grid line as per the rule of thirds. The background - skyline and lighthouse; now is two thirds of the photo, and the foreground -rocks; are one third of the photo. Overall, the photo is now better; and it draws the viewer to the lighthouse and beautiful skyline.
There are several additional rules of composition,
The rule of odds: this has to do with quantity in a photo, if your taking photos of apples, for example, better to have 1, 3, 5, and so forth, instead of even numbered.
The rule of space:This rule applies to where things are going in a photo or where the implication is, for example; a car moving should be shot as it enters your photo, with distance in front of it, this implies that the car is going somewhere. And, for people shots, if someone was simply laughing in the photo or looking in a direction in the photo, leave space in that direction.
To read additional 'composition rules': Diagonal lines, geometrical shapes, framing images,
symmetrical balance, asymmetrical balance, tonal balance, and color balance; browse the Web Articles.
Balance For Photo Composition