A few things you need to remember: the wider the aperture, the less the items surrounding your subject will be in focus, depending upon the lens you are using and the distance you are standing from your subject. When I first started learning photography, this made me crazy. I laughed when an instructor told me "don't worry, it will be like knowing your own name in a few months."
I spent some time in a field with horses for this week's Project 52 photos. Thankfully, everyone cooperated. The photo below was taken at a focal distance of 70mm, f/2.8, ISO 200, focusing in on the first horse. It is a good illustration of how an aperture, opened up to its largest setting, can help the subject stand out from the rest of the elements in the scene. Because it would have been much too bright for the time of day, I increased my speed to 1/1600s. I like the way these three horses formed somewhat of a line which angled itself on a hillside. This was right before all three came up to investigate my camera to see if it was edible.
Here is a photo taken during the summer of 2015 using the same practice. Billie and Piper can often be found playing and generally hanging out together. Piper is also a master at photo-bombing. I think they teach Border Collies this in "How To Be A Border Collie" school. I loved the way Billie looked at the moment and did not want to sacrifice the shot until Piper left it, so I opened up the aperture all the way. (Focal length 50 mm, f/1.8, ISO 200, 1/200s)
That's just what I did with the image below, which gave it a whole different feel. (Focal length 70mm, f/2.8, ISO 200, 1/1600s)