The rule of thirds is one of the most used and useful technique that involves composition. The human eye prefers when a subject is off center, this is because it improves the balance of colours, shapes, etc in a composition and is also more engaging, although there are times where an image is nicer when the rule of thirds is not respected.
How does it work?
Some cameras have the option of placing 4 lines (like shown above) "through" your screen to help you compose better photographs, if you have this its helpful to select that option. What you see through your lenses is divided into 9 boxes and 4 intersection points. The intersection point are what you want to focus on because generally any subject placed on them is made interesting to the human eye.
The horizontal lines are for the any distinctive near horizontal line, i.e. a branch, and the vertical ones for near vertical objects (in the picture's case the butterfly with the flower). You don't have to find a subject for both all the lines but putting your subject at one of the 4 intersection points helps a lot. Basically keep the subject off-center.
2.CROPPINGPeople seem to think that cropping is like digitally editing a photo and making it fake, but in reality all its doing is its taking an image and cutting bits of the SIDE off not bits IN the image, there is a big difference. Cropping will make your images way more interesting and involving by applying the rule of thirds when cropping an image.
The example below is a great way to practice your rule of thirds (mentally draw the lines). The walls in the background are in line with the upper horizontal line and the seagull is on two intersection points and also on the left vertical line, see it?