To utilize the depth of field provided by a lens,
consider increasing the subject-to-background distance.
The farther the in-focus subject is from the background,
the more out of focus the background will be.
Put lots of distance between the subject and the background.
For indoor shots, you might put the subject in front of a large, (clean) window.
The Columbine flower picture above utilized a window for the background.
How about a reason to use a narrow aperture setting? The snake picture above utilized a bright flash/narrow aperture setting to make the close subject's background turn completely black. This is possible even in bright daylight if your subject is close enough - as was the case in the sample picture. I employ this tip frequently in macro photography.
A slight revision of this technique works with more subject-to-camera distance but requires very dim lighting - or complete darkness. Try night photography outdoors! Flash shadows on the background are non-existent when using this technique.
A common photography mistake is to have competing lines going through the background of your subject. In the example above, the tree row cuts through the buck's antlers degrading the picture. Another common example is a bright skyline going through the back of a person's head. The high contrast line will be very distracting.
Continued ... Digital Photography Tips - The Background (Part 3)