These points are mapped by placing a virtual grid over a composition which consists of two evenly spaces vertical lines and two horizontal of the same. These lines then create a grid of nine equal squares.

Importantly, this grid also now creates four points of intersection.

The theory of the rule of thirds is that by positioning subjects either on the intersections or along the lines you will produce a more pleasing image that is balanced and has interest created from the position of the subjects and thus avoiding the 'bullseye' shots.

A good example of this can be seen when looking at landscapes where the horizon is usually placed on the bottom of the two horizontal lines and any predominant vertical points of interest such as perhaps rocks and trees are lined up with the verticals.

In the example here, you can see the horizon is aligned to the bottom guide as are the cows. The cow shed in on the bottom right intersection and the sun around the top left.

Studies have apparently shown (I don't have the source) that these intersecting points are where the eye goes first when looking at an image, not to the centre of the frame. Try it, take some time to look at a few pictures and see where you are drawn to.
Of course as with any rule of thumb, it is just a guide - a good one - but still a guide. I find 'bullyseye' shots, those with the subject sitting bang in the middle of the grid works extremely well when you have very symmetrical images where the composition is almost entirely about the subject itself.

Get out and experiment but the key to any composition is to take your time, look around you, look at points of view and really look at the scene before you even get your camera out of the bag.

Happy shooting!
Gareth