DOF - Depth Of Field
DOF is the area (plane) of the photo that is in sharp focus. For landscapes, we usually want a a very deep DOF, so that everything in it is in focus. For portraits, we usually like to have the person in focus, while the background is blurred.
So you have a plane that is in focus - and it can be either in the foreground (so the background is blurry) or in the background (so the foreground is blurry).

For the most part, this is achieved via a lens with a large aperture (that's the small f/stop number), but a longer focal length - i.e. zoom - increases the effect. Often people ask how to achieve with a small P&S camera, but unfortunately that is not something those little things are very good at.

If you have manual controls, use the biggest aperture. If you don't, try setting it to the portrait setting which makes the camera choose the biggest aperture it is capable of. Make sure there is some distance between you, the subject, and the background. You can also attempt it on Macro setting - that may work to a certain extent. If you don't know how to do all these things, READ YOUR MANUAL.

For further reading, check out this Wikipedia article: http://en.wikipedia.org/wiki/Depth_of_field