Simple question...complicated answer. What you are talking about is "depth of field." In particular, you are looking for shallow (also known as narrow) depth of field. That is when the range of subject matter that is in focus is narrow and the rest is out of focus.
Several factors help you get shallow depth of field. They include sensor size, aperture size, focal length, and subject distance. I'll try to de-geek those terms as I explain them.
The first term, sensor size, refers to the size of a digital camera's sensor. That's the part that sees the scene and translates it into a digital image. On a film camera, it's the piece of film. Point and shoot camera's have very small sensors. Small sensors have lots of depth of field, so it is hard to get a picture with the subject in focus and everything else blurry with a point and shoot digital camera. DSLR camera's have much larger sensors and so it is much easier to get that affect with a DSLR.
The next factor is the aperture size. The aperture is the hole in the lens that let's light in. It adjusts in size based on how much light is available (the less light, the wider it opens), how long the shutter is open (the longer the shutter is open, the smaller the aperture is), and how sensitive the camera is (that's the ISO setting). To get the shallowest depth of field, you want the widest possible aperture. Most cameras have a mode that let's you adjust the aperture while the camera automatically sets everything else. The mode is typically labeled "Av". Go to that mode and make your aperture as wide as possible. Confusingly, smaller aperture (also called f/stop) numbers mean a wider opening. So you want to set the aperture to the smallest number possible.
The focal length is the amount of zoom. The more you zoom in, the less depth-of-field you'll have. So if you want just the subject in focus, zoom in as close as possible and then back up.
Finally, how close your subject is to the things behind it has a big effect. If your subject is leaning up against a wall, it is very hard to have them in focus and the wall blurry. On the other hand, if they move forward a long way and you stay focused on them, the wall will be blurrier.
So to summarize, set your camera to the widest aperture (smallest f/stop number), use your longest zoom, and move your subject as far from the background as possible. If you really want shallow DOF, consider getting a DSLR. If you really, really want shallow DOF, get a DSLR with a larger sensor (also called "full frame") and a lens with a really big aperture.
Jose!
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...but I'm going to go see if it will work...maybe if I just learn all the features on the cameras I have already I won't need a new one, darn it. 
