It's called DOF or depth of field. Make the DOF shallow enough to contain only your subject and physics will do the rest. This is best achieved with large aperture sizes.
Set the lens to use maximum aperture size (smallest f/number). You may have to set the camera to aperture priority mode or manual mode. Position your subject as far away from the background as possible. Position yourself as close to your subject as possible. Focus, compose then shoot.
Now is the time to appreciate prime lenses, those that do not zoom. Only in prime lenses can you see aperture sizes like f/1.8, f/1.4, f/1.2, f/1.0, f/0.7.
If you're using a zoom lens, try zooming in. Longer focal lengths make DOF shallower too.
So how do you do this with a point-and-shoot camera? Nearly impossible. Without any control of the camera, you can try positioning your subject as far away as possible from the background with you as near as possible to the subject. The point-and-shoot is made to make everything as clear as possible so even if the mountains at the back are miles away, it will still stay relatively clear. It doesn't hurt to try though. If your camera has a portrait mode, try that. You can try shooting in relatively lower lighting conditions to force the camera to use the widest aperture but you may not see the blurred background at all.
Now you know!