Rule of Thirds

The rule of thirds is a very common photography technique and it's often the first one that is taught. It ensures that people are looking at a visually interesting photograph.
To create the rule of thirds, imagine a 3x3 grid on top of the photo you are taking. You should try to put the important part of the image where the points intersect. You can also place the focal point of your image in one of the "thirds" that are created by the grid, making it look more interesting than if everything was centered.

As you can see, the eye of the bee is at one of the intersections, and the bee and flowers are in the last two vertical thirds of the photo, making it more appealing than if it was centered.

With this photo, the eye of the puppy aligns with one of the intersections. This photo is also effective because the puppy is in the last vertical third, facing the empty space, making it look like the puppy has space to run, rather that being at the end of the frame. If the puppy was in the first third facing the same way, he would have nowhere to go, and it would look unappealing.