Your questions isn't very clear, and your "examples" don't help much, either. There are too many photos on that page, I don't know which one you're talking about.
So I now have to guess what you actually mean, and I THINK you are simply talking about a shallow depth of field. That is something you achieve via your CAMERA, not via "photoshop".
DOF - Depth Of Field
DOF is the area (plane) of the photo that is in sharp focus. For landscapes, we usually want 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).
To achieve a shallow DOF (for background blur and/or bokeh), you need the following (in order of importance):
1) longer focal lengths - the longer it is (i.e. more zoom), the stronger the effect becomes - so 200mm is much better than 35mm
2) distance - a subject that is CLOSE, a background that is further away.
3) a lens with a large aperture (f/1.8 or f/2.8 work best)
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 because of their VERY TINY sensors. Even the better bridge camera still have MUCH smaller sensors than a DSLR does. However, given enough zoom, many of them can still get there to a certain extent.
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.
For further reading, check out this Wikipedia article: http://en.wikipedia.org/wiki/Depth_of_field
And here is a website with really good examples: http://www.althephoto.com/concepts/selectivefocus.php