If you've drooled over food photography with shallow depth of field or portraits where the subject is in sharp focus but the background is blurred, you probably think they were shot with a digital SLR and an expensive lens and aren't possible with your point-and-shoot. And, for the most part, you would be correct.
Without getting into the technical details (you can go here for that), cameras with large sensors, such as digital SLRs, are able to create more background blur than the much smaller sensors in an average point-and-shoot. Along with that, you can get lenses that have very large apertures, which also contributes to getting really blurry backgrounds. Small sensors have greater depth of field and small cameras have short focal lengths, both of which work against you when trying to blur what's behind your subject.
So, yes, if you want super blurry backgrounds, you're going to want a camera with a larger sensor and a better lens. If that's not an option and all you have is a point-and-shoot with a zoom lens, you can still cut down on distracting backgrounds and get shallow depth of field with these steps.
Learn your lens' focusing range
If you don't understand apertures or focal lengths, that's fine because you really don't need to for this. What you should find out, though, is the focusing range for your camera. More specifically, how closely the lens can focus at its widest position (fully zoomed out) and its telephoto position (fully zoomed in).
For example, the photo at the top of this story was taken with the Canon PowerShot SX260 HS. The camera's lens can focus as close as 2 inches from a subject at its wide end and 3.3 feet with the lens zoomed in. This information is usually available in the specifications section of your camera's manual or on the manufacturer's Web site.
Use macro mode for close-ups
Your camera's macro mode is designed to let you focus on subjects as closely as possible. To get a shallow depth of field, you'll want to take advantage of this mode (usually designated by a flower icon) and get as close as you can for the shot that you want. The Nikon Coolpix S8200 I used for the photo above can focus as close as 0.4 inch from a subject. Macro mode is also where compacts tend to produce their sharpest photos.
Also, most newer cameras have auto modes with scene detection, so the camera will automatically switch into macro mode as soon as it detects a close subject.
For other subjects, zoom all the way in and, again, get as close as possible
The longer the focal length the shallower depth of field appears. This is the one advantage most point-and-shoots have because even low-end models have some zoom. Of course, the longer the lens, the better off you'll be, so if you have something like the Fujifilm FinePix HS30EXR I used for the photo to the right, you'll be able to get even more background blur.
Remember the focusing range from the earlier step? You'll need to know that for this so you know just how close you can get to your subject and have them be in focus. The closer you can get to your subject, the better off you'll be.
Put as much distance between your subject and the background as possible
This goes hand in hand with the previous step. The more distance there is between your subject and the background, the more it will be out of focus.
And, if possible, use the largest available aperture
Larger apertures on small-sensor cameras don't have as big an affect on depth of field as they do on a dSLR. Plus, point-and-shoots typically have their largest apertures available only at their shortest focal lengths and, again, you want to use the longest focal length possible. However, it does have some effect, so if you can set your aperture, set it for the largest available (lowest f-stop number). For example, the bird photo above was taken with Panasonic's Lumix FZ200, which has a 600mm lens that can be set to a large f2.8 aperture.
You can always fake it with software
There's not much more to say about this. There are several ways to blur the background with software -- from basic and free to pricey but advanced. Similarly, Sony's Cyber-shots that use its Exmor R sensors and Fujifilm's FinePix cameras with EXR CMOS sensors have special modes that use in-camera processing to blur backgrounds and produce good results.