Composition, a word most commonly used by the artists. What a common man knows but cannot comprehend.
What a photographer is taught from the day one. Compose your frame intact before you press the shutter. Use the gridlines in the viewfinder for assistance.
Divide your frame into 3 equal parts horizontally and vertically. Place your subject of importance at the intersection point. That’s where the eye falls at the first glance.

Balance your images on right and left. Keep the horizon at one third or two third of the frame. Do not cut the lines in the middle of the frame. All are implied rules. They rather have to be followed explicitly irrespective of the kind or style of photography being practiced.
Consider a frame as below contained inside a rectangle in the normal aspect ratio of 2:3.
The upper edge of the frame is called the suspension or if we compare it with nature its the sky. The lower edge of the frame is called the support or if we compare it with nature its the earth. 
The two sides form the balance. It can be compared to the regular weight balance. For a balanced frame it should have elements that can balance the frame. Simple is it.

In the below image the scenery is beautiful but the image has to be balanced on the right hand side.  Putting the boatman on the right hand side at the intersection point balances the image and the emphasis is given to the boatman (becomes the important subject in the picture).

The rule of thirds. How this is arrived at is a simple logic. Here is the logic.
Simply give 5 one rupee coin to your maid at home and ask her to divide it into 2 unequal parts. Irrespective of whether she calculates or not the instinct would be 2 and 3.
This is the psychology behind it or the human mental balance. This was just a story to substantiate the below theory of composition.
One of the best books to learn composition in photography.

If you divide 5 = 2+3 you are mentally balanced with the result. But 5= 1+4 though right theoretically but we feel little uncomfortable. Do I make sense here. This holds true for 99% of the population.
Typically the ratios that appeal to us are of the form
1/3 = .333
2/3 =  .666

2/5 = .4
3/5 = .6 and so on but revolves around the above concept.
Similarly we tend to look for these ratios our eyes tend to scan the portions in the photos where these ratios fall approximately. Thus the rule of third.Now let me take you through some mathematics and simple geometry drawings.

BC = ½ AB

C is the center; CD the radius cut the arc.  The ratio is obtained which substantiate for the rule of thirds.  
This explains why our eyes scan the photos at places where our otherwise conscious mind would not look for or simply put the rule of thirds. Place the objects at the meeting points of these lines.
Last but not the least. Rules are meant to be broken. But knowing what you are breaking makes a difference.