It works like this:
Imaginary lines are drawn dividing the image into thirds both horizontally and vertically. You place important elements of your composition where these lines intersect.
Using the Rule of Thirds helps produce nicely balanced movies. Also, as you have to position things relative to the edges of the frame it helps get rid of ' tiny subject surrounded by vast empty space' syndrome.

You do not want to place the person in the middle of the image.

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===The Rule of Thirds===
The Rule of Thirds was the creation of painters in the Renaissance. The aim of this style of composition is to provide a bit of background in the photo, to allow the photo to tell a story. Renaissance painters found that the eye doesn't rest on the center of a photograph.
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Figure 1.) The rule or thirds

Simply the rule of thirds states that object of attention should be placed at the intersection of the dividing lines. (Green dots) But you can also place the object along one of dividing lines.
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Figure 2.) Can also be expressed horizontally...
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Figure 3.) Horizontal example of the rule of thirds
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Figure 4.) ...or vertically
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Figure 5.) Vertical examples of the rule of thirds

What the whole aim of the rule of thirds is to create a photograph which captures attention. The rule of thirds does that by using human instinct. Most people's attention doesn't travel to the center of a photograph, it scans the edges. This is probably rooted in human instinct developed when man had to avoid saber-tooth tigers and the like. So if your attention doesn't go for the middle then why should your subject be there?
Also you aren't limited to just one subject. You can two or more. Then it's recommended that you place the main subject on one dividing line and a secondary subject on another.

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Figure 6.) Multiple subjects according to the rule of thirds

The ROT enhance the impression of action. If an object is traveling from left to right, then placing it on the left side would give it the impression that it going to slide into the right side.
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Figure 7.) "Sliding into the frame"

But this isn't an "etched in stone" kind for rule.
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Figure 8.) "Sliding out of the frame"

Also in landscapes The rule of thirds can give the pictures a bias. If your horizon is centered, people will wonder if it's a land shot or a sky shot. Placing the horizon at a line, then tells the viewer that this is a sky or land shot without having them to guess.
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Figure 9.) Sky at sunset shot.