I get this question quite often. There are two things you must do and two things you must have to get that dramatic professional blurred background look in photography. By the way, this look I am about to teach you has a name, It's called "bokeh". Why? I have know idea but, it doesn't matter anyway. So let's get to the meat of this post already! Here we go.
What you need:
(1) A decent size camera sensor. At least 1.7th of an inch or larger. So unfortunately, camera phones and tiny basic cameras aren't going to give good bokeh. Moving on...
(2) Aperture mode. This allows you to adjust how much distance will be in focus between you, your subject, and the infinite horizon (shown in camera as f numbers).
What you have to do:
(1) You need to get as close as possible to the subject / person you are photographing. You can do this by walking closer (obviously) or zooming in. I always recommend zooming in to at least 3x zoom for a basic camera or 50mm for an SLR as opposed to walking in closer because of lens distortion (enlarged noses problem).
(2) You need to switch to aperture mode and adjust the aperture to the lowest number available for your camera or lens. f/4 or lower. The lower the number, less will be in focus and more of the background will be blurred (more bokeh!). All SLR lenses and basic cameras have different aperture numbers.
At this point you may be asking yourself, can I adjust aperture on my basic point & shoot camera? Answer: Depends on your camera. If you can't adjust it, there is one trick that may help. You need to create a darker environment. The less light there is, the lower the aperture number will go automatically. If it is too dark though, your shutter speed will drop too and cause motion blur. The entire shot will be blurry. Not good! Now go experiment with aperture numbers and have fun!
The pic on the left is set to aperture f/2.0, the one the right f/5.6. Look at the watch band.