A few things to help explain it a bit further, to GP's good information...
Shallow depth of field is definitely affected by aperture as described above...but a few other factors also make a difference.
First, the larger the sensor, the shallower the depth of field. Which is why little P&S cameras with sensors smaller than a keyboard key struggle to produce this effect, and DSLRs with sensors many times that size can do it more easily. Full-frame DSLRs are the kings of shallow depth of field.
Another factor is distance...your distance from the subject, and the subject's distance from the background. The greater the distance, the more chance of blurring the background. Superzoom cameras can often get this effect if you stand far from a subject and zoom in optically to fill the frame...assuming the background is also farther from the subject, you can get more blur in the background. The more zoom you use, and the farther the subject from the background, the better chance of the background being outside the focus area, causing blur.
Also, note that the common term 'bokeh' doesn't actually mean 'background blur'...it's actually used to describe the 'quality' of the background blur...how aesthetically pleasing it is. The Japanese term has become synonymous with a blurred background as a result of shallow depth of field, but there are many examples of a blurred background that would be considered to have terrible bokeh. The best way to judge bokeh is by looking at specular highlights like lights, or bright sun glints...are they kind of choppy, not circular, not well blended with the background, not creamy, etc? That's bad bokeh. Are they perfectly round, smooth edged, blend smoothly with muted colors that blend almost like a watercolor with the rest of the background? That's good bokeh. A few quick examples from two lenses of mine...this one is a lens not known for having very good bokeh - I can get a blurred background, but it looks harsh, choppy, odd, not smooth:
Here's another blurred background, this one from a lens that is known for excellent bokeh:
You can see how small light spots in the background have turned into perfect circles, and muted colors that blend into a smooth creamy background...that's due to the optical quality of the glass and the 9-blade aperture.
Your camera will likely struggle to produce significant background blur. It would take as much optical zoom as you can muster, the widest aperture you can get (lowest F-stop number), and as much distance from the background as you can get your subject to be. With a larger sensored DSLR, a nice lens, and a wide-open aperture, the same effect can be achieved at close portrait distances to the subject and with the background just a few feet behind them.