The rule of thirds allows you to apply balance to your photographs as well as take interesting photos. As they say 3 is a magic number and it appears quite a lot through out photography.
Don't position a subject directly on one o the intersections as it can make your image appear a little off. The best thing is to place it slightly to the side as the building in the example above has been. It has been placed slightly to the right of one of the vertical lines, which draws your eye towards it.
It can be useful in landscape photography as above to place the horizon along on of the horizontal lines. It is helpful for balance but also helps you to separate the pictures foreground and mid ground from the horizon. Don't forget your tripod if you are shooting landscapes!
Most cameras allow you to overlay a grid onto your live view. This may help you when you with your composition. This will allow to place the objects/landscape that you are shooting within the grid.
If it doesn't come naturally to you don't worry keep practising and at the day rules are there to be broken so why not break them?