The on-line explanations of silk screening fail to explain the purpose of the screen itself. They mention that a template is placed over the screen to form the image. OK, but any template can do that without the need for the screen. So, what purpose does the screen serve? (Or, is it just a cool-sounding name)? 
-Jinx
Most of the designs I’ve seen would flop all around if they weren’t attached to a screen.
As I recall the “screen” is a thin open weave fabric with a plastic film attached to the upper side. The design or image is made by cuttimg away the film to form the image. A paint is squeegeed across the surface forcing paint through the screen onto the paper or card stock to produce the product. A new screen is prepared for each color desired in the finished product. I think that a photo process is also used to produce the screens. Stencils are tedius for large production runs where screens can be used on various materials. Tee shirts, for example, can be on the streets for sale shortly after a big game celebrating a teams victory. Usually they actually prepare for both teams to win but only print/screen shirts for the anticipated sure winner.
Hope that helps. 
This way, you can make the template once, and then just slop the pigment/dye/whatever onto the screen once per garment (or once per screen if there’s more than one color), instead of painstakingly painting each garment.
In a regular paper stencil, if you want to make a letter O, you have to leave a bit of paper at the top and bottom so the central part of the stencil doesn’t move around: (). Now if you glued the stencil to a screen, you could do away with the little ‘supporting’ bits because the screen would hold the stencil bits in position while allowing ink to pass through. So screens allow you to print closed loops without having to realign all of the centers each time.
Brossa, that’s the best explanation I’ve read. Beyond that, I guess I’d actually have to see someone in the process of making a silk screening.
I silkscreened a good bit when I was in junior high. I think the other advantages of the screen are that it makes an easy surface for the squeegee to pass over (the stencil lays under your screen) and that it seems to limit the amount of paint that actually ends up on the surface you’re screening onto.
In addition to the above responses, the screen enables you to spread the color evenly across the surface (the squeegee removes the surplus). It also limits the amount of paint that seeps under the edges of the stencil.
In my parents’ old silkscreening shop, the screen was the stencil. They would coat each screen with a layer of light-sensitive emulsion, and allow it to dry. The design to be put on a shirt is printed out on special vellum paper. The vellum is then aligned and attached to the dry screen, and the whole shebang is exposed to intense UV light. This can be accomplished by setting them out in the sun, but my parents’ shop had a special light table that accomplished the task more quickly.
The UV light reacts with the emulsion, and the emulsion hardens. Where the light is blocked by the stencil, the emulsion stays soft. The screen is run under water, and the unexposed parts of the emulsion wash away, leaving clear screen. The exposed emulsion stays. This creates nice, sharp edges.
Depending on how detailed you want the end product to be, you can use a tighter or looser weave screen. The screen allows for far more detail - gradients, for example - than a simple stencil would allow. With a nice tight weave, your final product can be detailed down to the pixel level.
A purist would tell you that stencilling is not silkscreening!
A stencil is a cutout design that blocks ink from reaching the surface you want to print on. The idea has been used for thousands of years - look for the hand silhouettes on cave walls. Certainly you can stick your stencil to a screen for convenience.
A traditional silkscreen is made photographically. Spread a light-sensitive emulsion on the screen, then use either a cutout stencil or a design printed on a transparency to block the light while exposed areas harden. The unexposed areas wash away so the ink can penetrate to the background. Screens made like this can last for a great number of prints.
Recently, some people have been painting the negative designs directly onto a screen, which is a lot simpler than the photo process. “Real” silkscreeners tend to consider this pretty useless, since the screen wears out very quickly.
And a linguist would tell you that neither of the root words, “silk” or “screen,” means either “photo” or “emulsion.”
Your definition of silkscreening requires a photographic process. Then what do you call the silkscreening that existed beforee the advent of photographic processes? The artists working in this format probably thought they were silkscreening. Saying they were doing “serigraph” won’t answer my question either, because “the word serigraph was formed from the Latin word seri (meaning silk) and the Greek word graphos (meaning to draw)”