As mentioned there are photos that I take when I am totally using the rule of thirds. Here are a few examples of the extreme use of the rule.
Then there are examples of when the use of the rule is a bit more subtle but still present.
In this photo you can see that the elk is not exactly in one third of the image. The back end of the elk starts in one of the thirds. Your eyes follow his body and move toward his huge antlers. The flow keeps you eye focused on the elk which in this case is the main subject. You will notice as you look at the elk your eye is still noticing the surroundings of the elk but your eye is not drawn away from the elk.
Here is an example where I have used the road starting in one third of the image to lead you into the rest of the image. The road leads you into the depth of the shot, ending in the final third of the image.
There are times I do not think about the rule so much and think more of eye flow. I may want to lead your eye to the back part of a photo or I may want to make sure your eyes capture the full essence of the shot. In these kinds of situations I think less about the rule of thirds. I compose based on how your eye will be drawn into the photo.
The angle and framing of this shot was intended to have your eye follow and see the entire bridge and its length.
You can see the top of the waterfall is nearly dead center in this image. I used the rock formations and flowing river to lead your eye to the top of the falls.
If you spend time reviewing much of my work you will find I like tight cropping, especially with people and animals. Each of us see photos in a different way. I tend to like tight crops and full frame. I dont know why...just do.
I hope these examples will help you as you compose and frame your photos. Try not to make the same mistake I do, try and not frame your photo too tight to start with. Give yourself some room to do your cropping in your software. Still compose well in the view finder. Just when you think...ah...that is the way I want it to look, zoom out just a hair. That will give you some room to finalize the desired look when you crop it.