Assuming you mean a compact/point&shoot camera with "normal", yes, it is possible. It is more difficult however than higher-end cameras for several reasons:
1) Most compact cameras have small sensors. The focal length of a "normal" lens scales accordingly, i.e. the focal length is very short. Lenses with short focal lengths tend to depict everything sharp, which works against blurring the background. (The same effect works great for "focus-free" cameras, like handphone cameras and many webcams.)
2) Some compact cameras may have lenses with a relatively small maximum aperture. This also works against blurring out-of-focus areas.
3) Some compact cameras may not allow you to control the aperture setting of the lens (everything is automatic), so you're at the mercy of the camera.
The most effective way to achieve the blurred background despite such limitations is probably to photograph close-ups/macros against a far away background (cliche example: flower against landscape).
For fast moving objects, you can also blur the background (albeit in a different way) by panning during the exposure.
Besides of those points mentioned by LittleWolf, if the camera you are referring to is indeed a compact/ultra-compact camera that does not allow you to control the aperture, you can try using "portrait mode" if not shooting macro. In this mode the camera will use a longer focal length and wider aperture, hence creating shallower depth of field and compressed background, resulting in a blurrer background.
Yep basically that's it. In the macro mode pic you can see that the background is blur while the tissue box is clear. The background is not very blur though, this might be due to the camera is automatically selecting the narrowest aperture, or simply a limitation of your camera (as LittleWolf mentioned, most compact cameras have very deep DOF due to the short focal length of the lens). In the portrait pic, you can try standing further away from the tissue box so that it is in the camera's focusing range (min focus distance should be given in your cemera's manual). You should then get a clear tissue box and a blur background just like the 1st pic.
If your pns doesn't have manual settings, you can 'trick' it to get a shallower dof by:
1. Don't shoot in too bright conditions (this forces the camera to open its apperture)
2. Use the macro mode and keep your subject as far from the background as possible
3. If possible, zoom in to the maximum