What, really, does stacking images do?  Does it make the image sharper, brighter, etc.? 

It removes the deleterious effects of "seeing", primarily. It removes noise in the image and thereby improves the signal-to-noise ratio.

Here's how it works. In a digital astrophoto, there will be good pixels (signal, points of the image that "count") -- the things you want -- and bad pixels (noise). Some of the noise, especially in long exposures, comes from the chip and electronics in the camera and this is essentially random. Some of the "noise" comes from bad seeing: Have you noticed while observing a bright star, the Moon, or a planet how the view seems to "wiggle"? Well, in an image this causes distortion.

If you make a whole lot of images in sequence (as with an AVI file from a webcam), then stacking them tends to "average out" the seeing effects. It also almost completely cancels the electronic noise. What is called a "dark frame" can be subtracted from the image to remove the rest of the noise, because the electronic noise from frame to frame appears in the same pixels (places in the image) even though from session to session it is more random.

After stacking, what remains is the best parts of the image (the good stuff). You can apply post-processing software to further enhance the image.

This animation will help to demonstrate:

This is made from about 45 individual stacked images, each of which was made from separate AVIs of about 600 frames spaced about 5 minutes apart. You'll note that some of the frames are much "sharper" than others. Each was processed in precisely the same way. So, the differences are due to "seeing" (atmospheric turbulence). Even stacking sometimes doesn't produce a good result, but it will always produce the best result by removing noise.

That's actually really cool how it does that.  I would like to try it, using the afocal technigue.  But, I have a dob, so the image won't always be centered exacly the same way.  Is there a stacking software that will let me "grab" the image, so I can move each new image of the moon I have over the current picture, so all the images of the moon in the same place?

Yes, Registax will do that up to a point. When you use Registax, the first thing you do after loading the AVI file (or a series of still images) is surround an "alignment  feature" in the image with a small box. The pixels in the box are the ones that will be used to determine the quality of the individual frames and the difference between them and the reference frame (the one that was showing when you applied the alignment box).

If an image drifts across the field of view, Registax will "track it" until the box bumps into the edge of the image.  Any  frame where that happens will be discarded from the stack.

So, if the alignment feature is far enough from the edge, if the drift is not  too  fast or  too  jerky, and if the drift rate is  low enough to allow for many frames to be included in the  stack, then unguided, untracked photography with a dob will work.

The best thing to do  is try  it. Registax is  free to download and use. There are several download sites. Just  Google it.

Photoshop has a way to remove image noise, should this be done before stacking.

I use PhotoShop very sparingly, and not often. I like its Unsharp Mask feature, but my lunar and planetary images don't often require it. For DSO images, PhotoShop has many helpful features including noise reduction. I generally process DSO images using MaxImDL and then save PhotoShop for the trickier stuff, like masking.

I should point out I am NOT a master at PhotoShop work ...

I reviewed MaxImDL site and from what I read it looks and acts just like PhotoShop Elements. So I sent a letter off to Adobe for their inputs, maybe a photoshop pro from their side could point out the differences.

Hap:

Always do your stacking before you do any kind of smoothing or noise removal.   The stacking will remove much of what you want removed anyway...and you don't really want to jack with processing each frame separately...trust me on that one.

MaxIm, on the whole, is nothing like Photoshop...elements or otherwise.   I use BOTH to process my images.  MaxIm would be considered a software that's a bit more advanced, with excellent data acquisition tools to boot...I wouldn't recommend it for simple webcamming and beginning imaging with regular camera, digital or otherwise.

Registax performs a LOT of good stuff...and like Jeff said, you don't really need Photoshop to improve on those images.  In fact, there are some powerful sharpening and smoothing tools built into that package...and it's free to boot

for the record.- regiscrap 4 will soon be out with multi-point alignment, how this will perform and handle rotation on DSO i have no idea yet-  though maxim and IP are great programs- theres a couple freebies out that do an excellant job stacking images- cripe, even if you stack 3 images, youll see an improvement over a single- just dont try to overprocess it with such limited exposure time--hey, anybody got a good joke??- nevermind, ill go back to the madhouse- LOL