Usually when you want to take nice portrait pictures, you will use a nice, fast lens, and adjust your aperture to get a blurry background.
The reality is that most people have a simple point and shoot camera that does not allow manual aperture settings, or they don't know how (or why) to change the settings.
Next time you are trying to take a portrait photo, try zooming in all the way and standing back a bit. This will create a blurry background and have a nice effect. When you do this, make sure to have steady hands and good lighting (this probably will not work well indoors unless you are using a tripod).