The photo provided with the text on it, naming the various pieces, can be a bit misleading.
First - the piece of trim along the top edge of the siding may not be the same style as what you have around your windows - and that's assuming your windows don't have a channel built into them to receive the siding to begin with.
Also - what the poster has called J-channel, in the corner of the wall and soffett, could actually be F-channel, and there may or may not be a J-channel where the soffett meets the fascia. J and F channels are nailed in place usually. I've done a lot of work on vinyl sided homes, and there is another type of trim that is popular along the top edge of the siding - its actually a two piece channel system, that works very much like the trim at the bottom, which holds the vinyl skirting in place. The back piece is nailed to the wall, the siding is run up, and the cover piece of trim then snaps into the channel that was nailed in place.
Now - to replace a screen on a window - any modern window - you should NOT have to remove any of the siding, as the screen sets inside the window frame. Most modern screens have pull tabs, or D-rings at the bottom, with which you can grip, and pull up on the screen, to remove it from it's channel in the window. There are 2 springs in the upper corner of the screens that provide the tension to hold the screen in place. Modern screens are made to be easy to push out, in an emergency. If the rings or tabs a the bottom of the screen are missing (fairly common), a putty knife works as a good tool to easily pry the screen up enough to get the bottom of it out of the channel.
If you actually needed to replace the whole window, then you would want to remove the siding surrounding the window. This is done by pulling down on the bottom of the siding, so that it unsnaps from the row below it....then you simply pull the nails or staples holding it in place. You CAN purchase a 'siding tool' from any hardware, or building supply store, that will save your fingers some wear and tear - the tool only costs a few dollars, and is well worth the purchase. When you go to replace the siding, use roofing nails (I like at least an inch in length, to get my fingers under the nail head, heheh) and do NOT hammer the nails in tight! The siding needs to be able to slip side to side due to expansion and contraction. For the same reason, put your nail in the center of the slot, not to one side. If you nail the siding too tight, or get your nails off center, the siding will bow, or look wavy, as it gets pinched, or stretched, by the nailing pattern.
If you need - feel free to send me a photo of YOUR window, that you are trying to replace the screen on, and I may be able to point out the easiest way to fix it. Be happy to help.
Have Fun
Link to siding tool -
http://www.magnumtools.com/shopexd.asp?id=5147