A partial extraction is when you only allow one portion of an image to remain intact, and cut out the background on the rest of the image. Photos that work well for this are ones where the main subject is extruding somehow--somebody waving a flag, waving hi, or even something like the head of your pet. This obviously really highlights the extruding part, so unless you're focussing your layout on your dog's wagging tail, the hind end might not be your best bet for a partial extraction. (Ha, ha ha ha. Badaboom.)
To do the one in the following layout about the awesome Arum dracunculus growing in front of our Post Office, I first used my rectangular lasso to choose the part of the photo I wanted to keep, as the part that I extracted had the sidewalk as a background, with a bit of litter--just not a great photo overall, background-wise. I copied that square, then pasted it on a different layer.
Next, I went back to the base layer and started extracting (my favorite tool is the pen, working very close up; complete your path, switch to the path palette, click on the "change path to selection" little circle of dots at the bottom, feather 2 pixels, invert the selection, and press delete to remove the background).
I chose to break my extraction about 1/3 of the way into the square, so that part of the extraction overlapped part of the original. When I added a very strong shadow to the square (100% using a dark color eyedrop-picked from the background paper), I duplicated that for the extraction. Of course, you'll get a shadown down the middle of your square that clearly delineates the extraction, so you have to fix that...
Right-click on the layer name for the extraction, and pick "create layer"--this will make your style effects into separate layers, 1 for each style you have. Here it was just the shadow, so I made all my changes to that one.
Working only on the newly-created shadow layer, I first erased the strong line that showed where the extraction began--but only the vertical part of the shadow. In fact, I extended the shadow under the spathe (the petal-like structure of the plant) further to the left, gradually fading it out, using both the blur and the smudge tools. This really helped the spathe pop out dimensionally against the background, but the effect was subtle enough that you wouldn't really notice it unless you knew, I think...
Please click on the image for credits and a little more info about this gorgeous Dragon, aka Stink Lily!