Where using thirds makes for a really dynamic image, using a centred composition makes for a static image. This can be cool too! There can be a certain calmness to a centred image that we love. But there are a few guidelines we use to pull it off...
The main situation a centred composition works well in is when you have one single isolated subject, and lots of symmetry in the image to keep your eye from wandering around. Remember the lone tree example from the straight horizon lines lesson? That is one of those single isolated subjects that having a centred composition worked beautifully with!
If there had been another horse or even some sheep scattered around in the foreground of the above photo, the eye wouldn't have known what to focus on and would have skipped around between all the possible subjects. Instead, having just the one single point of focus means you can use a nice symmetrical composition and there is no question about what to look at. Even when you are using a centred composition vertically in the frame, you can still use thirds horizontally. The hills in the photo above take up one third of the frame and the sky takes up two thirds.
This cute little bed head is as centred as it gets! Even though this image is a little more cluttered than the one above, there is still a sense of symmetry, the fireplace frames her, and those crazy tinsel horns add to the sense of symmetry as well.
Above is another more chaotic example of using a centred composition. That funny mermaid is dead centre of the frame, and even though the room is an explosion of kid stuff and life stuff, the symmetry in the window framing her head and the leading lines in the couch draw your eye right to her and make a sense of balance and order amongst the chaos.
Ahhhh, how peaceful is this little centred scene? So much balance and symmetry, and that lovely light pouring in and then falling out of the frame. Kate could have walked into the hallway more and cut out that door on the left, but then there wouldn't have been that lovely light falloff creating that beautiful golden to dark gradient on that crazy 50's carpet.