1. FIND A SIMPLE BACKGROUNDThere's a reason why portrait studios have roll upon roll of neutral backdrops—they don't distract from the subject and they're relatively easy to light. Most of us don't have the room for a dedicated photo studio, so we have to be more creative.
For a studio look at home, a blank wall, closed window curtains or even a bed sheet taped to the ceiling all can provide a clean backdrop for your image. You can use a single light placed behind your subject to illuminate the background and create a sense of space and depth. If you're working with on-camera flash, try bouncing it off of the ceiling for softer, more diffused light.
When shooting outdoors, look for a large wall or a garage door (alleys at midday are a good place to start), or an open vista where you can use the sky as your background.
2. REDUCE YOUR DEPTH OF FIELDUsing a large aperture for shallow depth of field not only will help create a pleasing, soft background, but it can let you be creative with out-of-focus foreground elements, too.
Be careful, though! With a shallow depth of field, it's especially critical to be sure that your subject's facial features are in focus, so try manual focusing. Review images on your camera's LCD preview to zoom in on facial details and check that they're sharp.
Or consider exposure bracketing with multiple apertures; it's safer than finding that your favorite shots are slightly out of focus when you view them on a larger screen. Though a pleasing background is important, focus is fundamental.