Once again, aperture is the hardest leg of the exposure triangle to understand. Long story short: Large apertures tend to put the background out of focus; Small apertures tend to have more of the scene in focus.
The flower on the left was shot with a large aperture, f/4. The one on the right was shot with a small aperture, f/22. In this case, most people would agree that the photo on the left is more pleasing.
Here is a beautiful landscape photograph that uses a very small aperture. Notice how the front of the dock, closest to the camera, is sharp.
And here is a portrait of President Obama that uses a very large aperture:
Notice how his eyes are sharp but his ears are blurry.
Here is a gorgeous portrait of a swimmer. The photographer has masterfully set the aperture to a value that is not too large and not too small, so that the pool in the background is visible, but blurry. (Her use of light is also masterful!)